Mirror Symmetry and Tropical Geometry

CONTEMPORARYMATHEMATICS

American Mathematical Society

527

Mirror Symmetry and Tropical Geometry

NSF-CBMS Conference onTropical Geometry and Mirror Symmetry

December 13–17, 2008Kansas State University

Manhattan, Kansas

Ricardo Castaño-BernardYan Soibelman

Ilia ZharkovEditors


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American Mathematical SocietyProvidence, Rhode Island

CONTEMPORARYMATHEMATICS

527


NSF-CBMS Conference on Tropical Geometry and Mirror Symmetry

December 13–17, 2008 Kansas State University

Manhattan, Kansas

Ricardo Castaño-Bernard Yan Soibelman

Ilia Zharkov Editors

Editorial Borad

Dennis DeTurck, managing editor

George Andrews Abel Klein Martin J. Strauss

2000 Mathematics Subject Classification. Primary 14J32, 14T05, 53D37, 53D40, 14N35,32S35, 58A14, 14M25, 52B70.

Library of Congress Cataloging-in-Publication Data

NSF-CBMS Conference on Tropical Geometry and Mirror Symmetry (2008 : Kansas State Uni-versity)

Mirror symmetry and tropical geometry : NSF-CBMS Conference on Tropical Geometry andMirror Symmetry, December 13–17, 2008, Kansas State University, Manhattan, Kansas / RicardoCastano-Bernard, Yan Soibelman, Ilia Zharkov, editors.

p. cm. — (Contemporary mathematics ; v. 527)Includes bibliographical references.ISBN 978-0-8218-4884-5 (alk. paper)1. Tropical geometry—Congresses. 2. Calabi-Yau manifolds—Congresses. 3. Algebraic

varieties—Congresses. 4. Symmetry (Mathematics)—Congresses. 5. Mirror symmetry—Congresses. I. Castano-Bernard, Ricardo, 1972– II. Soibelman, Yan S. III. Zharkov, Ilia,1971– IV. Title.QA582.N74 2010516.3′5—dc22 2010017905

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10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10

Contents

Preface vii

Invited Lectures ix

Closed form expressions for Hodge numbers of complete intersectionCalabi-Yau threefolds in toric varieties

Charles F. Doran and Andrey Y. Novoseltsev 1

Anchored Lagrangian submanifolds and their Floer theoryKenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono 15

Motivic Donaldson-Thomas invariants: summary of resultsMaxim Kontsevich and Yan Soibelman 55

On the structure of supersymmetric T 3 fibrationsDavid R. Morrison 91

Log Hodge groups on a toric Calabi-Yau degenerationHelge Ruddat 113

Tropical theta characteristicsIlia Zharkov 165

v


Preface

Mirror symmetry has sparked enormous interest and revolutionized variousareas of mathematics—notably, algebraic and symplectic geometry. This has trig-gered the development of powerful new techniques to understand problems whichare important for both string theory and mathematics. Kontsevich’s HomologicalMirror Symmetry provided a conjectural explanation in terms of triangulated cat-egories associated with mirror dual Calabi-Yau manifolds. It was complementedby Strominger, Yau and Zaslow’s more geometric conjecture, suggesting that mir-ror symmetry could be understood in terms of dual special Lagrangian fibrations.Later, the work of Kontsevich and Soibelman, Gross and Wilson led to view theSYZ conjecture as a limiting statement in which the underlying structures con-trolling mirror symmetry are integral affine manifolds and piecewise linear objects,which are now called tropical varieties. Tropical geometry has recently started torelease its full power to understand the involution of geometric structures posed bymirror symmetry. This tropicalization of mirror symmetry is a promising approachthat aims at remarkably simplifying problems depending on various parameters ina non-linear way to much simpler combinatorial problems.

The conference covered a variety of topics related to tropical geometry and mir-ror symmetry. Mark Gross’ lectures, devoted to his program with Bernd Siebert,will be published separately by the AMS as a volume of the CBMS MonographSeries. Some of the contributions of other participants are included in the presentvolume. They illustrate vast connections of mirror symmetry and tropical geometrywith other areas of mathematics and mathematical physics. The techniques andmethods used by the authors of the volume are at the frontier of this very activearea of research. The reader will benefit from Dave Morrison’s insightful surveyof the evolution and future of the SYZ conjecture. With the flavor of the classicalBatyrev and Borisov construction, Doran and Novoseltsev provide an algorithm tocompute string-theoretic Hodge numbers for the intersection of two hypersurfacesin 5-dimensional toric varieties. Using tropical degeneration data and logarithmicgeometry, Ruddat gives a version of mirror duality for ordinary Hodge numbers,stringy Hodge numbers and the affine Hodge numbers. On the symplectic side,Fukaya, Oh, Ohta and Ono provide a version of Lagrangian Floer homology foranchored Lagrangians and its (higher) product structures; this article may serve asa partial outline for their book Lagrangian Intersection Floer Theory: Anomaly andobstruction, volume 46, parts I & II, AMS/IP Studies in Advanced Math., 2009.Kontsevich and Soibelman present a cohesive picture of the study of the categor-ical, geometric, and formal computational aspects of Donaldson-Thomas theory.Zharkov studies tropical counterpart of the classical theory of theta characteristics.

vii

viii PREFACE

We expect young researchers will find in this volume a variety of unsolved problemsto think about.

Acknowledgements: The CBMS Conference on Tropical Geometry & Mirror Sym-metry, Manhattan KS, 2008 was sponsored by NSF grant 0735319. The preparationof this volume was partially supported by NSF 0735319 and NSF-FRG 0854989.

R. Castano-Bernard, Y. Soibelman, I. Zharkov, Editors

Invited lectures

• Mohammed Abouzaid (M.I.T.): “Homological Mirror Symmetry for T 4

and applications to Lagrangian embeddings”.• Kwok Wai Chan (Harvard): “SYZ transformations and mirror symme-try”. Abstract: In joint work with Conan Leung, we propose a programwhich is aimed at understanding mirror symmetry by using Fourier-typetransformations (SYZ transformations). In this talk, we will discuss theapplications of SYZ transformations to mirror symmetry for toric Fanomanifolds. In particular, we will see how quantum cohomology is trans-formed to Jacobian ring and how Lagrangian torus fibers are transformedto matrix factorizations.

• Charles Doran (U of Alberta): “Algebraic Cycles, Regulator Periods, andLocal Mirror Symmetry”.

• Kenji Fukaya (Kyoto University): “Singularity theory over Novikov ringand Mirror symmetry” (joint with Oh-Ohta-Ono). Abstract: Generatingfunction of open-closed Gromov-Witten invariant that is potential func-tion with bulk in our sense, is a formal function which is some kinds offormal power series converging in appropriate adic topology, over univer-sal Novikov ring. In the case of toric manifold and its Lagrangian fiberthis function provides a nice example of universal family of hypersurfacesingularities, and becomes a ‘rigid analytic analogue’ of K. Saito’s theoryof isolated hypersurface singularity. This is actually a global theory andso is different from classical Saito’s theory. Saito’s theory is an imortantsource of so called Frobenius manifold structure (= Saito’s flat structure).Another important source of Frobenius manifold structure is (big) qua-tum cohomology. We find that they coincides in the case of arbitrary toricmanifold.

• Ilia Itenberg (U of Strasbourg): “Welschinger invariants of toric Del Pezzosurfaces” (joint work with V. Kharlamov and E. Shustin) Abstract: TheWelschinger invariants are designed to bound from below the number ofreal rational curves passing through a given generic real collection of pointson a real rational surface. In some cases these invariants can be calculatedusing G. Mikhalkin’s approach which deals with a corresponding count oftropical curves. Using the tropical approach we establish a logarithmicequivalence of Welschinger and Gromov-Witten invariants in the case ofgeneric collections of real points on a toric Del Pezzo surface equippedwith an arbitrary real structure (with non-empty real part).

ix

x INVITED LECTURES

• Ludmi Katzarkov (U of Miami): “Conic Bundles Old and New”. Ab-stract: We will formulate a HMS approach to a classical question of ra-tionality of conic bundles.

• David Morrison (University of California, SB): “SYZ and the moduli ofCalabi–Yau threefolds.”

• Anvar Mavlyutov (Oklahoma State U): “Deformation of toric varietiesand Calabi-Yau hypersurfaces”. Abstract: In the 90’s, Klaus Altmannstudied deformations of affine toric varieties. He constructed families ofdeformations of affine toric varieties as complete intersections in anothertoric variety using Minkowski sums of polyhedra. We found a generaliza-tion of this construction for arbitrary toric varities. In a particular impor-tant case of complete simplicial toric varieties which are partial crepantresolutions of the projective toric varieties corresponding to reflexive poly-topes, this new construction coincides with our previous construction ofdeformations of such toric varieties obtained by a different method via ho-mogeneous coordinates. These deformations are important as they inducedeformations of Calabi-Yau hypersurfaces.

• Yong-Geun Oh (U of Wisconsin-Madison): “Seidel’s exact sequence forclosed Calabi-Yau manifolds”. Abstract: In this talk, we will explain howconstruction of Seidel’s long exact sequence of Floer cohomology underthe symplectic Dehn twists can be extended to general, especially closed,Calabi-Yau manifolds. The highlight of the talk is our usage of the notionof ‘anchored Lagrangian submanifolds’ and some study of compactnessissue of the moduli space of pseudo-holomorphic sections in the setting ofsymplectc Lefschetz fibrations.

• Tony Pantev (U of Pennsylvania): “Mirror symmetry for del Pezzo sur-faces”. Abstract: I will discuss the general mirror symmetry question fordel Pezzo surfaces in a setup that goes beyond the Hori-Vafa ansatz. Iwill describe the mirror map explicitly and will describe non-trivial testsfor homological mirror symmetry. This is a joint work with Auroux,Katzarkov and Orlov.

• Bernd Siebert (U of Hamburg): “The tropical vertex”. Abstract: Oneinsight of mirror symmetry is the fact that the enumerative geometry ofrational curves is related to the deformation theory of a “mirror vari-ety”. Now there is a pro-nilpotent group of automorphisms of the alge-braic 2-torus ruling the constructions of maximal degenerations (Kontse-vich/Soibelman, Gross/S.). On the mirror side this group should havesome enumerative geometry meaning. In the talk I will present joint workwith M. Gross (UCSD) and R. Pandharipande (Princeton) showing thatthis group indeed organizes a class of natural enumerative geometry prob-lems on toric surfaces into an algebraic structure. The correspondenceruns via tropical geometry.

• Yan Soibelman (Kansas State U): “Complex integrable systems, stabilitystructures and invariants of Donaldson-Thomas type”. Abstract: In a re-cent joint work with Maxim Kontsevich we offered a general approach toDonaldson-Thomas type invariants (“counting of BPS states” in the lan-guage of physicists). I am going to discuss an application of our approachto complex integrable systems of Seiberg-Witten type. In particular, I will

INVITED LECTURES xi

explain how the wall-crossing formulas in Seiberg-Witten theory are re-lated to the wall-crossing formulas of our DT-invariants, and how tropicalgeometry appears in the description of the spectrum of SW-model.

• Benjamin Young (McGill University): “Counting colored 3D Young di-agrams with vertex operators”. Abstract: I will show how to computesome multivariate generating functions for 3D Young diagrams (otherwiseknown as “plane partitions”). Each box in a 3D Young diagram gets as-signed a “color” according to a certain pattern; the variables keep track ofhow many boxes of each color there are. My generating functions turn outto be orbifold Donaldson-Thomas partition functions for C3/G, where Gis a finite abelian subgroup of SO(3). If time permits, I will discuss recentwork on the more general problem of the orbifold topological vertex andthe combinatorial crepant resolution conjecture.


Contemporary Mathematics

Closed Form Expressions for Hodge Numbersof Complete Intersection Calabi-Yau Threefolds

in Toric Varieties

Charles F. Doran and Andrey Y. Novoseltsev

Abstract. We use Batyrev-Borisov’s formula for the generating function ofstringy Hodge numbers of Calabi-Yau varieties realized as complete inter-sections in toric varieties in order to get closed form expressions for Hodgenumbers of Calabi-Yau threefolds in five-dimensional ambient spaces. Theseexpressions involve counts of lattice points on faces of associated Cayley poly-topes. Using the same techniques, similar expressions may be obtained forhigher dimensional varieties realized as complete intersections of two hyper-

surfaces.1

1. Introduction

In [1] Batyrev obtained combinatorial formulas for Hodge numbers h1,1(X) andhn−1,1(X) of an n-dimensional Calabi-Yau variety X arising as a hypersurface in atoric variety associated to a reflexive polytope. It is immediate from these formulasthat h1,1(X) = hn−1,1(X◦), where X◦ is Batyrev’s mirror of X, and this equalitysuffices to show that mirror symmetry holds on the level of Hodge numbers forCalabi-Yau 3-folds. However, it is also important to consider higher dimensionalCalabi-Yau varieties including singular ones.

Batyrev and Dais, motivated by “physicists Hodge numbers”, introduced string-theoretic Hodge numbers [5] for a certain class of singular varieties. The string-theoretic Hodge numbers coincide with the regular ones for smooth varieties andwith regular Hodge numbers of a crepant desingularization if it exists. LaterBatyrev also introduced stringy Hodge numbers [2] for a different class of singularvarieties. While stringy and string-theoretic Hodge numbers are not the same, theydo agree for the varieties we will be dealing with in this paper, see [7] for furtherdetails on relations between them.

Batyrev and Borisov were able to obtain a formula for the generating func-tion of string-theoretic Hodge numbers in the case of complete intersections in toric

1In fact, at the time of publication of this article authors have obtained such formulas, theywill be communicated in a subsequent work.

2010 Mathematics Subject Classification. Primary 14J32.The first author was supported in part by NSF Grant No. DMS-0821725 and NSERC-SAPIN

Grant #371661.

c©2009 Charles F. Doran, Andrey Y. Novoseltsev

1

Contemporary MathematicsVolume 527, 2010

1

2 CHARLES F. DORAN AND ANDREY Y. NOVOSELTSEV

varieties and show that this function has properties corresponding to mirror symme-try [3]. While their formula can be used in practice for computing Hodge numbers(as it is done in software PALP [13]), it is recursive, takes significant time evenon computers, and does not provide much qualitative information on particularHodge numbers. This work was motivated by the desire to obtain, for completeintersections, formulas similar to those for hypersurfaces.

We were able to accomplish this goal in the case of two hypersurfaces intersect-ing in a five dimensional toric variety, see Theorem 4.1 for arbitrary nef partitionsand a simplified expression in Theorem 4.6 for the indecomposable ones. The algo-rithm allows one to get expressions for h1,1 for the intersection of two hypersurfacesin a higher dimensional ambient space as well.

Acknowledgements. We would like to thank Victor Batyrev, MaximilianKreuzer, Anvar Mavlyutov, John Morgan, and Raman Sanyal for inspiring discus-sions and references. We are grateful to our referee for his or her thorough reviewof our paper and pointing out quite a few possible improvements as well as typos.

It was also very beneficial for this project to be able to experiment with nu-merous examples using lattice polytope module [14] of the software systemSage [15], which provides convenient access to PALP [13] as one of its features.

2. Generating functions for stringy Hodge numbers

In this section we fix the notation, define a nef partition and the generatingfunction for the stringy Hodge numbers of the associated variety. The exposition isbased on [3, 6], where one can also find further properties of the objects in question(the notation there is slightly different, as those authors work with faces of cones,not of supporting polytopes). Since this paper is mostly combinatorial, we will usethe generating function to define the stringy Hodge numbers.

Let N � Zn be a lattice of dimension n, M = Hom(N,Z) be its dual lattice,NR = N ⊗Z R and MR = M ⊗Z R be the vector spaces spanned by these lattices.Let Δ ⊂ NR be a reflexive polytope (a bounded convex polytope of dimension nwith vertices at lattice points and all facets having integral distance 1 from theorigin) and Δ◦ ⊂ MR be its polar (the convex hull of inner normals of facets of Δ,normalized to be primitive integral vectors — reflexivity of Δ implies that Δ◦ isalso reflexive, hence the name). Let PΔ be the toric variety corresponding to thefan spanned by faces of (a triangulation of the boundary of) Δ, see, for example, [9]for details on constructing PΔ.

Let the vertex set V = V (Δ) be partitioned into a disjoint union of subsets

V = V1

∐V2

∐. . .

∐Vr ,

with corresponding polytopes Δi = Conv(Vi, 0). This decomposition determines anef partition if the Minkowski sum Δ1 + · · ·+Δr is also a reflexive polytope whichwe will denote by ∇◦, for ∇ ⊂ MR.

We say that this nef partition is indecomposable, if the Minkowski sum of anyproper subset of {Δi}ri=1 is not a reflexive polytope in the generated sublattice. De-composable nef partitions correspond to products of Calabi-Yau varieties, presentedas complete intersections of smaller numbers of hypersurfaces in toric varieties ofsmaller dimensions.

2

HODGE NUMBERS OF CICY THREEFOLDS IN TORIC VARIETIES 3

The associated Cayley polytope P ∗ ⊂ NR × Rr of dimension n+ r − 1 is givenby

P ∗ = Conv(Δ1 × e1,Δ2 × e2, . . . ,Δr × er) ,

where {ei}ri=1 is the standard basis for Zr ⊂ Rr. The Cayley polytope supportsthe Cayley cone C∗ ⊂ NR × R

r of dimension n + r. It is a reflexive Gorensteincone of index r (rP ∗ is a reflexive polytope) with dual Cayley cone C ⊂ MR × Rr

supported on the dual Cayley polytope P .The intersections of P with affine subspaces given by intersections of hyper-

planes xi = 1, xj = 0 for a fixed i ∈ {n+1, . . . , n+r} and all j ∈ {n+ 1, . . . , n+ r},j �= i, are polytopes ∇1, . . . ,∇r corresponding to the dual nef partition such that∇ = Conv(∇1, . . . ,∇r) and Δ◦ = ∇1 + · · · + ∇r. These polytopes determineequations of hypersurfaces in PΔ and their intersection is a (possibly decomposableand/or singular) Calabi-Yau variety X of dimension n− r.

Faces of the cone C with the inclusion relation form an Eulerian poset withthe minimal element the vertex at the origin and the maximal element C itself.It is convenient to use faces of P to index elements of this poset, with ∅ and Prepresenting the vertex of C and C itself respectively. If x is a face of P , we willdenote by x∨ the dual face of P ∗.

For any Eulerian poset P with the minimum element 0 and the maximumelement 1, if x, y ∈ P with x � y we will use notation rkx for the rank of x, thelength of the longest chain of element between 0 and x, [x, y] = {z ∈ P : x � z � y}for the subposet of elements between x and y inclusively, and dP = rk 1 for therank of the poset.

If x and y are faces of P with x ⊂ y, then d[x, y] = dim y − dimx, rkx =dim x+ 1, and it is natural to define here dim ∅ = −1, since dimensions of faces ofP are less by one than the dimensions of corresponding faces of C.

Definition 2.1. Let P be an Eulerian poset of rank d with the minimal element0 and the maximal one 1. For d = 0 let GP = HP = BP = 1. For d > 0 definepolynomials GP , HP(t) ∈ Z[t] and BP(u, v) ∈ Z[u, v] recursively by

HP(t) =∑

0<x�1

(t− 1)rkx−1G[x,1](t),

GP(t) = τ<d/2(1− t)HP(t),

where

τ<d/2

∞∑k=0

aktk =

∑0�m<d/2

aktk

is the truncation operator, and∑

0�x�1

B[0,x](u, v)ud−rkxG[x,1](u

−1v) = GP(uv).

Proposition 2.2. The BP polynomial defined above has the following proper-ties:

(1) The degree of BP(u, v) in v is (strictly) less than dP/2.(2) If dP � 2, then BP(u, v) = (1− u)dP .(3) If P is Eulerian poset associated to faces of a polygon with k vertices/edges,

then dP = 3 and BP(u, v) = 1 + [k − (k − 3)v](u2 − u)− u3.

3


Proof. See [3], Examples 2.8, 2.9, and Proposition 2.10. �

Definition 2.3. Let F ∈ N be a d-dimensional lattice polytope (or a d-dimensional face of a lattice polytope). Let �(F) = |F ∩N | be the number oflattice points inside F . Let �∗(F) be the number of points in the relative interiorof F . (For a point both � and �∗ are equal to 1.) Define functions SF and TF by

SF (t) = (1− t)d+1∞∑k=0

�(kF)tk,

TF (t) = (1− t)d+1∞∑k=1

�∗(kF)tk.

We also set S∅ = 1.

Proposition 2.4. For F �= ∅ the functions SF and TF defined above have thefollowing properties:

(1) SF (t) = t1+dTF (t−1).

(2) SF (t) = 1 + [�(F)− d− 1]t+ higher order terms.(3) TF (t) = �∗(F)t+ [�∗(2 · F)− (d+ 1)�∗(F)]t2 + higher order terms.(4) SF is a polynomial of degree at most d.(5) TF is a polynomial of degree exactly d+ 1.(6) SF has degree d− r + 1 and SF (t) = td−r+1SF (t

−1) if and only if F is aGorenstein polytope of index r.

Proof. For 1 see [3], Proposition 3.6 and references there. The next twoproperties are immediate from the definition. Then 4 and 5 follow from 1–3. For 6see [6], Remark 2.15 and references there. �

Definition 2.5. The generating function associated to the dual Cayley coneC of a nef partition is

EC(u, v) =1

(uv)r

∑∅�x�y�P

(−1)1+dimxu1+dim ySx

(vu

)Sy∨(uv)B[x,y]

(u−1, v

),

and its coefficients are the stringy Hodge numbers of the Calabi-Yau variety X upto a sign:

EC(u, v) =∑p,q

(−1)p+qhp,q(X)upvq .

Remark 2.6. The formula above is taken from [12], the original one in [3] isless convenient for actual computations since it includes infinite sums. A similarformula is also given in [6] (line 11 on page 57), but there is a typo — the posetsof B-polynomials must be dualized.

Remark 2.7. It is not obvious from the expression for EC that it is a polyno-mial of degree 2(n−r), although this is so for C coming from a nef partition. On theother hand, the definition of EC makes sense for any (n+r)-dimensional Gorensteincone of index r and it is conjectured that it is always such a polynomial [6].

4


3. The hypersurface case

In this section we will derive a formula for h1,1 of a hypersurface Calabi-Yauthreefold in the four-dimensional toric variety PΔ using the generating function.While this formula can be obtained by other means, [1], it will serve as motivationand demonstration of techniques that will be used for nef partitions in the nextsection.

Theorem 3.1. Let Δ be a four-dimensional reflexive polytope. Let X ⊂ PΔ bea generic anticanonical Calabi-Yau hypersurface. Then

h1,1(X) = �(Δ)− 5−∑

dim y=0

�∗(y∨) +∑

dim y=1

�∗(y) · �∗(y∨),(3.1)

where each sum runs over the faces of Δ◦ of the indicated dimensions.

Proof. A hypersurface can be treated in the above framework as a completeintersection with r = 1, P ∗ � Δ, and P � Δ◦.

The generating function for n = 4 and r = 1 is given as

uvEC(u, v) =∑

∅�x�y�P

(−1)1+dim xu1+dim ySx

(vu

)Sy∨(uv)B[x,y]

(u−1, v

),

so h1,1(X) = h2,2(X) is equal to the coefficient of u2v2 or u3v3. Below, by exten-sively using Propositions 2.2 and 2.4 without further mention, we will determine thecoefficient of u3v3 in the term corresponding to each pair (x, y) on the right handside. The reason for concentrating on a “high v-degree” term is that it allows usto deal only with simple B-polynomials corresponding to Eulerian posets of smallrank, as we will see below. Note also that for the current case dim y∨ = 3− dim yand d[∅, P ] = 5.

First of all, observe that terms with B depending on v do not contribute tothe coefficient of u3v3. Indeed, if the v-degree of B is positive, then d[x, y] � 3and we must have dim x � 1, dim y � 2, i.e. dim y∨ � 1, and at least one of theseinequalities is strict. Then either both S-polynomials are equal to one or one isequal to one and the other is linear. On the other hand, B[x,y]

(u−1, v

)could only

have v-degree 2 or more if d[x, y] � 5, which is only possible for [x, y] = [∅, P ],where both S-polynomials are equal to one. Therefore, the product of all thesepolynomials does not contain a u3v3 term.

Next we are going to consider cases with d[x, y] � 2 and either x = ∅ ory = P . The reason for separating these cases from the rest is that the degree ofS∅ = SP∨ = 1 is not bounded by dim ∅ = −1.

Suppose x = ∅ and dim y � 1. Then the corresponding term of the generatingfunction is

u1+dim ySy∨(uv)(1− u−1)1+dim y = Sy∨(uv)(u− 1)1+dim y,

where Sy∨ is a polynomial of degree at most 3 − dim y. We see that in the onlypossible cases for dim y = −1, 0 the contribution to u3v3-term is determined by thethird degree term in Sy∨ and, by using the symmetry property of the S-polynomialof the reflexive polytope P ∗, we obtain

�(P ∗)− 5,(3.2)

−∑

dim y=0

�∗(y∨).(3.3)

5


Suppose dimx � 2 and y = P . Then the corresponding term of the generat-ing function is

(−1)1+dimxu5Sx

( vu

)(1− u−1)4−dimx,

where Sx is a polynomial of degree at most dimx, which must be 3 or 4 in order tohave any term with v3. In these cases the contribution is determined by the third

degree term in Sx, however,(vu

)3

must be multiplied by u6 in order to get u3v3,

which is not possible. We see that there are no contributions to u3v3-term.We consider remaining cases, d[x, y] = 0, 1, 2, with x �= ∅ and y �= P .Suppose x �= ∅, y �= P , and x = y. Then the corresponding term of the

generating function is

(−u)1+dim ySy

(vu

)Sy∨(uv),

where Sy and Sy∨ are polynomials of degrees at most dim y and 3−dim y. In orderto get v3, we need to multiply the leading terms of these polynomials. The u-degreeof the v3 term in the total product will be 1+dim y−dim y+3−dim y = 4−dim y.Since we are interested in u3 terms, we must have dim y = 1. The correspondingcontribution is ∑

dim y=1

�∗(y) · �∗(y∨).(3.4)

Suppose x �= ∅, y �= P , and dim y = 1+dimx or dim y = 2+dimx. Then wesee that there are no contributions to h1,1(X), since the total degree of the productof the S-polynomials is at most 2.

Now combining all contributions we obtain (3.1), which completes the proof.�

Remark 3.2. The terms of (3.1) have the following meaning. Torus-invariantdivisors of the ambient space, corresponding to lattice points of Δ, except forthe origin, have 4 linear relations between them. Divisors corresponding to theinterior points of the facets do not intersect a generic Calabi-Yau hypersurface,while divisors corresponding to the interior points of faces of codimension two maybecome reducible when intersected with this hypersurface.

Corollary 3.3. If h1,1 = 1 for a Calabi-Yau hypersurface in the toric varietyassociated to a 4-dimensional reflexive polytope Δ, then Δ is a simplex.

Proof. This easily follows from (3.1), if we split �(Δ) into the sum of internalpoints of all of its faces:

h1,1(X) = �∗(Δ) +∑

dim y=0,1,2,3

�∗(y∨)− 5−∑

dim y=0

�∗(y∨) +∑

dim y=1

�∗(y) · �∗(y∨)

=∑

dim y=1,2

�∗(y∨) +

⎡⎣ ∑dim y=3

�∗(y∨)− 4

⎤⎦+

∑dim y=1

�∗(y) · �∗(y∨).

Since faces dual to faces of dimension 3 are vertices of Δ, we see that the term inbrackets is positive while all others are non-negative, and if h1,1(X) = 1, Δ musthave exactly 5 vertices, i.e. be a simplex. �

6


Remark 3.4. While the number of reflexive polytopes of any fixed dimensionis finite (up to GL(Z) action) and there is an algorithm allowing one to constructall of them (realized in PALP [13]), this number for dimension 5 and higher is sobig, that it is practically impossible. However, results similar to the above corollarycan allow for construction of all reflexive polytopes corresponding to Calabi-Yauvarieties with small Hodge numbers.

4. The bipartite complete intersection case

In this section we derive the closed form expression for h1,1(X) of a bipartiteCalabi-Yau threefold complete intersection in the five-dimensional toric variety PΔ.

Theorem 4.1. Let Δ be a five-dimensional reflexive polytope. Let X ⊂ PΔ bea complete intersection Calabi-Yau threefold corresponding to a fixed nef partitionof Δ with associated dual Cayley polytope P . Then

h1,1(X)=�(P ∗)− 7 −∑

dim y=0

[�∗(2 · y∨)− 6 · �∗(y∨)]

+∑

dim y=1

�∗(y∨) +∑

dim y=1

�∗(y) · [�∗(2 · y∨)− 5 · �∗(y∨)]

−∑

dim y=2

[�(y)− �∗(y)− 3] · �∗(y∨) −∑

dim x=2dim y=3x<y

�∗(x) · �∗(y∨)

+∑

dim y=3

[�∗(2 · y)− 4 · �∗(y)] · �∗(y∨),

where sums run over faces of P of indicated dimensions.

Proof. In this case we have the following relation for the generating function:

(uv)2EC(u, v) =∑

∅�x�y�P


(vu

)Sy∨(uv)B[x,y]

(u−1, v

),

so h1,1(X) = h2,2(X) is equal to the coefficient of u3v3 or u4v4. Below we willdetermine the coefficient of u4v4 in the term corresponding to each pair (x, y) onthe right hand side. Note, that dim y∨ = 5− dim y.

First of all, let’s consider all (x, y)-pairs with positive v-degree of B[x,y]. Sinced[∅, P ] = 7, the highest possible v-degree of B[x,y] is 3. However, this is the onlypair when degree 3 is a possibility and S∅ = SP∨ = 1, thus it does not give acontribution to u4v4. If the v-degree of B[x,y] is 2, then d[x, y] � 5 and either bothS-polynomials are equal to 1, or one of them is 1 and the other is linear, so againsuch pairs yield no contribution to u4v4. If the v-degree of B is 1, then d[x, y] � 3and it is easy to see that only for x = ∅ and dim y = 2 or dim x = 3 and y = P itis possible to have the total v-degree of Sx and Sy∨ greater than 2.

Suppose x = ∅ and dim y = 2. Let k(y) be the number of vertices of y. Thenthe corresponding term of the generating function is

u3Sy∨(uv)(1 + [k(y)− (k(y)− 3)v](u−2 − u−1)− u−3)

= Sy∨(uv)(u3 + [k(y)− (k(y)− 3)v](u− u2)− 1),

7


where Sy∨ is a polynomial of degree at most 3. Its leading coefficient is �∗(y∨),thus the contribution to h1,1 is

−∑

dim y=2

(k(y)− 3)�∗(y∨).(4.1)

Suppose dimx = 3 and y = P . Let k(x) be the number of vertices of x∨.Then the corresponding term of the generating function is

u7Sx

(vu

)(1 + [k(x)− (k(x)− 3)v](u−2 − u−1)− u−3)

= Sx(uv)(u7 + [k(x)− (k(x)− 3)v](u5 − u6)− u4),

where Sx is a polynomial of degree at most 3. We see that terms with v4 haveu-degree 2 or 3, thus there is no contribution into h1,1.

Next we are going to consider cases with d[x, y] � 2 and either x = ∅ ory = P . The reason for separating these cases from the rest is that the degree ofS∅ = SP∨ = 1 is not bounded by dim ∅ = −1.

Suppose x = ∅ and dim y � 1. Then the corresponding term of the generatingfunction is

u1+dim ySy∨(uv)(1− u−1)1+dim y = Sy∨(uv)(u− 1)1+dim y,

where Sy∨ is a polynomial of degree at most 5−dim y. We see that in all three casesfor dim y = −1, 0, 1 the contribution to h1,1 is determined by the fourth degree termin Sy∨ and, using that P ∗ is a six-dimensional Gorenstein polytope of index 2, weobtain

�(P ∗)− 7,(4.2)

−∑

dim y=0

[�∗(2 · y∨)− 6 · �∗(y∨)] ,(4.3)

∑dim y=1

�∗(y∨).(4.4)

Suppose dimx � 4 and y = P . Then the corresponding term of the generat-ing function is

(−1)1+dimxu7Sx

( vu

)(1− u−1)6−dimx,

where Sx is a polynomial of degree at most dimx. We see that in all three casesfor dimx = 4, 5, 6 the contribution to h1,1 is determined by the fourth degree term

in Sx, however( vu

)4

must be multiplied by u8 in order to get u4v4. We see that

this is not possible and there are no contributions to h1,1.We consider remaining cases, d[x, y] = 0, 1, 2, with x �= ∅ and y �= P .Suppose x �= ∅, y �= P , and x = y. Then the corresponding term of the

generating function is

(−u)1+dim ySy

(vu

)Sy∨(uv),

where Sy and Sy∨ are polynomials of degrees at most dim y and 5−dim y. Considerdegree α term in Sy, 0 � α � dim y. In order to get v4 we need to multiply it bydegree 4 − α term from Sy∨ , 0 � 4 − α � 5 − dim y. Then the u-degree of thisproduct with u1+dim y will be 1+ dim y−α+4−α = 5+dim y− 2α. Since we are

8


interested in u4 terms, possible values for (dim y, α) satisfying all the restrictionsare (1,1) and (3,2). The corresponding contributions are

∑dim y=1

�∗(y) · [�∗(2 · y∨)− 5 · �∗(y∨)] ,(4.5)

∑dim y=3

[�∗(2 · y)− 4 · �∗(y)] · �∗(y∨).(4.6)

Suppose x �= ∅, y �= P , and dim y = 1+dim x. Then the corresponding termof the generating function is


(vu

)Sy∨(uv)(1− u−1)=(−u)1+dim xSx

(vu

)Sy∨(uv)(u− 1),

where Sx and Sy∨ are polynomials of degrees at most dim x and 4 − dimx, thusonly the product of their leading terms yields terms with v4. Taking into accountremaining factors of the product, we see that possible u-degrees of terms with v4 are1+dimx−dimx+4−dimx = 5−dim x and greater by one, 6−dimx. Therefore,we get u4v4 terms and contributions to h1,1 only for dimx = 1 or dim x = 2:

−∑

dimx=1,dim y=2,x<y

�∗(x) · �∗(y∨),(4.7)

−∑

dimx=2,dim y=3,x<y

�∗(x) · �∗(y∨).(4.8)

Suppose x �= ∅, y �= P , and dim y = 2 + dim x. Then we see that there areno contributions to h1,1, since the total degree of S-polynomials is at most 3.

Now let’s combine all the contributions:

h1,1=�(P ∗)− 7 −∑

dim y=0

[�∗(2 · y∨)− 6 · �∗(y∨)]

+∑

dim y=1

�∗(y∨) −∑

dim y=2

(k(y)− 3)�∗(y∨)

+∑

dim y=1

�∗(y) · [�∗(2 · y∨)− 5 · �∗(y∨)]−∑

dim x=1dim y=2x<y

�∗(x) · �∗(y∨)

−∑

dim x=2dim y=3x<y

�∗(x) · �∗(y∨) +∑

dim y=3

[�∗(2 · y)− 4 · �∗(y)] · �∗(y∨).

Observe, that terms (4.1) and (4.7) can be naturally combined, since the first onecontains the number of vertices of a 2-face y, while the second one sums over internalpoints of all edges of each 2-face y. Therefore, the total contribution of these twoterms is

−∑

dim y=2

[�(y)− �∗(y)− 3] · �∗(y∨),

where �(y) − �∗(y) is the number of boundary points of y. This leads us to thestated formula for h1,1(X) and completes the proof. �

9


Lemma 4.2. In the notation of Theorem 4.1, we have

h3,3(X) = 1 +∑

dim y=1

�∗(y) · �∗(y∨)−∑

dim y=0

�∗(y∨),

h2,3(X) =∑

dim y=2

�∗(y) · �∗(y∨),

h3,2(X) = −∑

dim y=2

[�(y) + 3 · �∗(y)− 3− �∗(2 · y)] �∗(y∨).

Proof. The same type of argument as for the proof of the theorem (butshorter). �

Corollary 4.3. In the notation of Theorem 4.1, if the nef partition is inde-composable, the following relations hold:

∑dim y=1

�∗(y) · �∗(y∨) =∑

dim y=0

�∗(y∨),(4.9)

∑dim y=2

�∗(y) · �∗(y∨) = 0,(4.10)

∑dim y=2

�∗(2 · y) · �∗(y∨) =∑

dim y=2

[�(y)− 3] �∗(y∨).(4.11)

Proof. Follows immediately from Lemma 4.2, since we know that h3,3(X) = 1and h2,3(X) = h3,2(X) = 0. �

Now we use this corollary to prove the following result.

Lemma 4.4. Let Δ be a five-dimensional reflexive polytope. Let P be the dualCayley polytope of an indecomposable two part nef partition of Δ. If y is a face ofP , then �∗(y) · �∗(y∨) = 0.

Proof. First, let y be a vertex. Then �∗(y) = 1 and we need to show that�∗(y∨) = 0. Note that y∨ is a 5-dimensional facet of P ∗. Then either y∨ is one ofthe polytopes ∇1 or ∇2 of the dual nef partition and it does not have an interiorpoint, since the nef partition is indecomposable ([6], Corollary 6.12), or y∨ has non-empty intersection with both ∇1 and ∇2. In the latter case consider the projectionof NR ×R2 ⊃ P ∗ onto the second factor. Then the image of y∨ is the line segmentfrom (1, 0) to (0, 1), which does not have interior points. Therefore, in any case�∗(y∨) = 0 as desired.

Now from relations (4.9) and (4.10) we conclude the result for dim y � 2, butthen for dim y � 3 it follows by symmetry. �

Remark 4.5. Relation (4.11) follows from Lemma 4.4 and Pick’s formula.Indeed, let y be a face of P of dimension 2, such that �∗(y∨) �= 0, then we knowthat �∗(y) = 0. Then the area of y is A(y) = �(y)/2− 1 and

A(2 · y) = �∗(2 · y) + �(2 · y)− �∗(2 · y)2

− 1 = 4A(y) = 2�(y)− 4,

but the number of boundary points of 2 · y is 2�(y), thus �∗(2 · y) = �(y)− 3.

10


Theorem 4.6. Let Δ be a five-dimensional reflexive polytope. Let X ⊂ PΔ be acomplete intersection Calabi-Yau threefold corresponding to a fixed indecomposablenef partition of Δ with associated dual Cayley polytope P . Then

h1,1(X) = �(P ∗)− 7−∑

dim y=0

�∗(2 · y∨) +∑

dim y=1

�∗(y∨)

+∑

dim y=1

�∗(y) · �∗(2 · y∨) −∑

dim x=2dim y=3x<y

�∗(x) · �∗(y∨)

−∑

dim y=2

�∗(2 · y) · �∗(y∨) +∑

dim y=3

�∗(2 · y) · �∗(y∨).

where sums run over faces of P of indicated dimensions.

Proof. Follows from Theorem 4.1 and Lemma 4.4. �

5. Relations with other results

It would be desirable to have a geometric interpretation for each term of theobtained expressions for Hodge numbers and, in particular, to be able to identifythe toric component of h1,1(X), given by images of the toric-invariant divisors ofthe ambient space or, equivalently, the polynomial part of h2,1(X), correspondingto polynomial deformations of the complete intersection in the ambient space. (Inthe hypersurface case this extra information follows “for free” from the proof ofBatyrev’s formulas for the Hodge numbers.) While there is an algorithm for com-puting the toric part of the cohomology ring (see [8], for example), it does not givedirectly a “closed form” expression for its dimension. Also Borisov and Mavlyutovhave constructed complete stringy cohomology spaces in [7] for semiample hyper-surfaces in toric varieties and perhaps their techniques may be used in completeintersection case as well.

It would also be interesting to compare the result of Theorem 4.6 with thepreviously known formulas for Hodge numbers of complete intersections obtainedby Batyrev and Borisov in [4]. They have considered a special case when all divisorscorresponding to the nef partition in the non-resolved variety (i.e. the varietycorresponding to Δ without triangulation of the boundary) are ample.1 Theirformulas restricted to our case are given below, although we were not yet able tomatch all terms with ours.

Definition 5.1. A lattice polytope Δ′ is a Minkowski summand of anotherlattice polytope Δ if there exist μ ∈ Z>0 and a lattice polytope Δ′′ such thatμΔ = Δ′ +Δ′′.

If the divisors given by polytopes ∇i are ample in PΔ (before partial resolutioncorresponding to a triangulation of ∂Δ), then Δ◦ is a Minkowski summand of ∇i

for all i, all these polytopes are combinatorially equivalent, and each face θ ofΔ◦ decomposes into Minkowski sum θ =

∑i θi, where θi is a face of ∇i of the

1In [10] the first author and John Morgan relate the closed form expressions of [4] directly tothe (mixed) Hodge structure on middle-dimensional cohomology for complete intersection Calabi-Yau threefolds in toric varieties. A generalization that allows for geometric interpretations foreach term and identification of the toric components of h1,1(X) would be most useful for suchapplications.

11


same dimension as θ. In this case, the nef-partition is necessarily irreducible andTheorem 4.6 is applicable. Another way to compute Hodge numbers in this case isthe following result.

Corollary 5.2 (from Corollary 8.4[4]). Let Δ be a five-dimensional reflexivepolytope. Let X ⊂ PΔ be a complete intersection Calabi-Yau threefold correspondingto a fixed nef partition of Δ with ample divisors corresponding to polytopes ∇i ofthe dual nef partition. Then

h1,1(X) = �(Δ)− 6−∑

dim θ=4

�∗(θ)−∑

dim θ=3

�∗(θ)

+∑

dim θ=2

�∗(θ) ·[�∗(θ∗)− �∗(θ∗1)− �∗(θ∗2)

],

h2,1(X) =[�∗(2∇1 +∇2)− �∗(2∇1) + �∗(∇1 + 2∇2)− �∗(2∇2)

]− 7

−∑

dim θ=0

[�∗(θ∗)− �∗(θ∗1)− �∗(θ∗2)

]

+∑

dim θ=1

�∗(θ) ·[�∗(θ∗)− �∗(θ∗1)− �∗(θ∗2)

],

where the sums are over faces of Δ of indicated dimensions, θ∗ is the face of Δ◦

dual to θ, and θ∗ = θ∗1 + θ∗2 is the decomposition into Minkowski sum with θ∗i beinga face of ∇i.

Proof. Follows from Corollary 8.4 in [4], after restricting to dimΔ = 5 anddimX = 3. �

6. Examples

In this section we apply our formula to several simplices and non-simplices Δwith the corresponding h1,1(X) equal and not equal to 1. The reflexive polytopesconsidered here were taken from the data supplements to [11].2

Example 6.1 (Simplex, h1,1(X) = 1). Let vertices of Δ be given by columnsof the matrix

⎛⎜⎜⎜⎜⎝

0 −3 0 1 0 00 −3 1 0 0 00 −2 0 0 0 10 −2 0 0 1 01 −1 0 0 0 0

⎞⎟⎟⎟⎟⎠

and consider the nef partition corresponding to V1 = {1, 3, 5} and V2 = {2, 4, 6}.Applying Theorem 4.6, we get (each sum is written as a separate term)

h1,1(X) = 8− 7− 0 + 0 + 0− 0− 0 + 0 = 1,

h2,1(X) = 98− 7− 30 + 0 + 0− 0− 0 + 0 = 61.

2Available at http://hep.itp.tuwien.ac.at/∼kreuzer/CY/hep-th/0410018.html

12


Example 6.2 (Non-simplex, h1,1(X) = 1). Let vertices of Δ be given bycolumns of the matrix ⎛

⎜⎜⎜⎜⎝

1 1 −1 0 0 0 11 −3 0 1 0 0 00 −1 0 0 1 0 01 −2 0 0 0 1 02 −2 0 0 0 0 0

⎞⎟⎟⎟⎟⎠

and consider the nef partition corresponding to V1 = {1, 2, 5, 6} and V2 = {3, 4, 7}.Applying Theorem 4.6, we get (each sum is written as a separate term)

h1,1(X) = 9− 7− 1 + 0 + 0− 0− 0 + 0 = 1,

h2,1(X) = 54− 7− 14 + 0 + 5− 0− 1 + 0 = 37.

Example 6.3 (Non-simplex, h1,1(X) �= 1). Let vertices of Δ be given bycolumns of the matrix ⎛

⎜⎜⎜⎜⎝

−1 1 0 0 0 1 −2 −1−1 0 1 0 0 0 −1 00 1 0 1 0 0 −1 0−1 1 0 0 1 0 −1 0−2 2 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎠

and consider the nef partition corresponding to V1 = {1, 3, 7} and V2 = {2, 4, 5, 6, 8}.Applying Theorem 4.6, we get (each sum is written as a separate term)

h1,1(X) = 10− 7− 1 + 0 + 0− 0− 0 + 0 = 2,

h2,1(X) = 46− 7− 11 + 0 + 1− 0− 0 + 1 = 30.

Example 6.4 (Simplex, h1,1(X) �= 1). Let vertices of Δ be given by columnsof the matrix ⎛

⎜⎜⎜⎜⎝

−1 0 0 0 1 0−1 0 1 0 0 0−1 0 0 1 0 0−2 1 0 0 0 1−2 2 0 0 0 0

⎞⎟⎟⎟⎟⎠

and consider the nef partition corresponding to V1 = {2, 6} and V2 = {1, 3, 4, 5}.Applying Theorem 4.6, we get (each sum is written as a separate term)

h1,1(X) = 9− 7− 0 + 0 + 0− 0− 0 + 0 = 2,

h2,1(X) = 83− 7− 27 + 0 + 10− 0− 1 + 0 = 58.

Remark 6.5. As can be seen from these and a few other examples, none of theterms in our formula vanishes identically.

References

[1] Victor V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces intoric varieties, J. Algebraic Geom. 3 (1994), no. 3, 493–535, arXiv:alg-geom/9310003v1.MR MR1269718 (95c:14046)

[2] , Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Inte-grable systems and algebraic geometry (Kobe/Kyoto, 1997), World Sci. Publ., River Edge,NJ, 1998, pp. 1–32, arXiv:alg-geom/9711008v2. MR MR1672108 (2001a:14039)

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[3] Victor V. Batyrev and Lev A. Borisov, Mirror duality and string-theoretic Hodge numbers,Invent. Math. 126 (1996), no. 1, 183–203, arXiv:alg-geom/9402002v1. MR MR1408560(97k:14039)

[4] , On Calabi-Yau complete intersections in toric varieties, Higher-dimensional complexvarieties (Trento, 1994), de Gruyter, Berlin, 1996, pp. 39–65, arXiv:alg-geom/9412017v1.MR MR1463173 (98j:14052)

[5] Victor V. Batyrev and Dimitrios I. Dais, Strong McKay correspondence, string-

theoretic Hodge numbers and mirror symmetry, Topology 35 (1996), no. 4, 901–929,arXiv:alg-geom/9410001v2. MR MR1404917 (97e:14023)

[6] Victor V. Batyrev and Benjamin Nill, Combinatorial aspects of mirror symmetry, Integerpoints in polyhedra—geometry, number theory, representation theory, algebra, optimization,statistics, Contemp. Math., vol. 452, Amer. Math. Soc., Providence, RI, 2008, pp. 35–66,arXiv:math/0703456v2 [math.CO]. MR MR2405763 (2009m:14059)

[7] Lev A. Borisov and Anvar R. Mavlyutov, String cohomology of Calabi-Yau hypersurfaces viamirror symmetry, Adv. Math. 180 (2003), no. 1, 355–390, arXiv:math/0109096v1 [math.AG].MR MR2019228 (2005b:32056)

[8] Volker Braun, Burt A. Ovrut, Maximilian Kreuzer, and Emanuel Scheidegger, Worldsheetinstantons and torsion curves. B. Mirror symmetry, J. High Energy Phys. (2007), no. 10,023, 53, arXiv:0704.0449v1 [hep-th]. MR MR2357955 (2009i:14079)

[9] David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, MathematicalSurveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999.MR MR1677117 (2000d:14048)

[10] Charles F. Doran and John W. Morgan, Algebraic topology of Calabi-Yau threefoldsin toric varieties, Geom. Topol. 11 (2007), 597–642, arXiv:math/0605074v1 [math.AG].MR MR2302498 (2008i:32033)

[11] Albrecht Klemm, Maximilian Kreuzer, Erwin Riegler, and Emanuel Scheidegger, Topolog-ical string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections,J. High Energy Phys. (2005), no. 5, 023, 116 pp. (electronic), arXiv:hep-th/0410018v2.MR MR2155395 (2006h:81263)

[12] Maximilian Kreuzer, Erwin Riegler, and David A. Sahakyan, Toric complete intersections and

weighted projective space, J. Geom. Phys. 46 (2003), no. 2, 159–173, arXiv:math/0103214v3[math.AG]. MR MR1967149 (2004h:14044)

[13] Maximilian Kreuzer and Harald Skarke, PALP: a package for analysing lattice polytopeswith applications to toric geometry, Comput. Phys. Comm. 157 (2004), no. 1, 87–106,arXiv:math/0204356v1 [math.NA]. MR MR2033673 (2004j:52014)

[14] Andrey Y. Novoseltsev, lattice polytope module of Sage, The Sage Development Team,2009, http://sagemath.org/doc/reference/sage/geometry/lattice polytope.

[15] William A. Stein et al., Sage Mathematics Software (Version 3.3), The Sage DevelopmentTeam, 2009, http://www.sagemath.org.

Department of Mathematical and Statistical Sciences, 632 CAB, University of Al-

berta, Edmonton, Alberta T6G 2G1, Canada

E-mail address: [email protected]

Department of Mathematical and Statistical Sciences, 632 CAB, University of Al-

berta, Edmonton, Alberta T6G 2G1, Canada


14

Anchored Lagrangian submanifoldsand their Floer theory

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

Abstract. We introduce the notion of (graded) anchored Lagrangian sub-manifolds and use it to study the filtration of Floer’ s chain complex. We thenobtain an anchored version of Lagrangian Floer homology and its (higher)product structures. They are somewhat different from the more standard non-anchored version. The anchored version discussed in this paper is more nat-urally related to the variational picture of Lagrangian Floer theory and so tothe likes of spectral invariants. We also discuss rationality of Lagrangian sub-manifold and reduction of the coefficient ring of Lagrangian Floer cohomologyof thereof.

Contents

1. Introduction2. Novikov rings3. Anchors and abstract index4. Anchors, action functional and action spectrum5. Grading and filtration6. Orientation7. Floer chain complex8. Obstruction and A∞ structure9. Comparison between anchored and non-anchored versions10. Reduction of the coefficient ring and Galois symmetryReferences

Key words and phrases. Floer homology, anchored Lagrangian submanifolds, Novikov ring,Fukaya category, (BS-)rational Lagrangian submanifolds, N-rationalization.

KF is supported partially by JSPS Grant-in-Aid for Scientific Research No.18104001 andGlobal COE Program G08, YO by US NSF grant # 0503954, HO by JSPS Grant-in-Aid forScientific Research No.19340017, and KO by JSPS Grant-in-Aid for Scientific Research, Nos.18340014 and 21244002.

1


c©2010 American Mathematical Society

15

1619212325293435444853

2 K. FUKAYA, Y.-G. OH, H. OHTA, K. ONO

1. Introduction

Lagrangian Floer theory associates to each given pair of Lagrangian submani-folds L0, L1 ⊂ M a group HF (L1, L0), called the Floer cohomology group. Floercohomology group can be regarded as a (∞/2-dimensional) homology group of thespace of paths Ω(L0, L1) joining L0 to L1:

(1.1) Ω(L0, L1) = {� : [0, 1] → M | �(0) ∈ L0, �(1) ∈ L1}.Floer [Fl1] used Morse theory to rigorously define this cohomology group. Theexterior derivative of the ‘Morse function’ Floer used is the action one-form αdefined by

(1.2) α(�)(ξ) =

∫ 1

0

ω(�(t), ξ(t)) dt

for each tangent vector ξ ∈ T�Ω(L0, L1).In general the one form α is closed but not necessarily exact. So one needs

to use Novikov’s Morse theory [N] of closed one forms. In order to take care ofnon-compactness of the moduli space of connecting orbits which occurs from non-exactness of the closed one form involved, Novikov uses a kind of formal powerseries ring, the so called Novikov ring for his Morse theory of closed one forms.

Floer, and later Hofer-Salamon [HS] and the fourth named author [On], useda similar Novikov ring for Floer homology of periodic Hamiltonian system. Thepresent authors also used a Novikov ring to study Lagrangian Floer homology in[FOOO00]. They however introduced a slightly different ring which they calluniversal Novikov ring. The same universal Novikov ring was used in [Fu2] toassociate a filtered A∞ category (Fukaya category) to a symplectic manifold, whichcombine Floer cohomologies of various pairs of Lagrangian submanifolds, togetherwith their (higher) product structures.

In Section 5.1 [FOOO09], the relationship between the Floer cohomology overa (traditional) Novikov ring and the one over the universal Novikov ring is discussed,which concerns pairs of Lagrangian submanifolds. The discussion thereof involvesa systematic choice of associating base points on the connected components ofΩ(L0, L1) when the pair (L0, L1) varies. In this paper we extend this to the casesof three or more Lagrangian submanifolds, which enter in the product structure ofFloer cohomology.

We remark that the closed one form α above, determines a single-valued func-tion on an appropriate covering space of Ω(L0, L1) up to addition of a constant.The choice of this additive constant, which is closely related to the choice of a basepoint, determines the filtration of Floer cohomology. When more than two La-grangian submanifolds are involved, to equip filtrations of the Floer cohomologies‘in a consistent way’ for all pairs (L0, L1) is a somewhat nontrivial problem. Theproblem of finding a systematic choice of the base point for the filtration sharessome similarity with the corresponding problems for the degree (dimension) andfor the orientation of the moduli space of pseudo-holomorphic strips or polygons.(See Definition 3.7.)

For the purpose of systematically finding the base points of the path spaces,we use the notion of anchored Lagrangian submanifold.

Definition 1.1 (Anchored Lagrangian submanifolds). Fix a base point y ∈ M .An anchor of a Lagrangian submanifold L ⊂ M to y is a path γ : [0, 1] → M

16

ANCHORED LAGRANGIAN SUBMANIFOLDS 3

such that γ(0) = y, γ(1) ∈ L. A pair (L, γ) is called an anchored Lagrangiansubmanifold.

Roughly speaking, Floer cohomology group is a cohomology group of a chaincomplex CF (L1, L0) which is generated by the set of intersections L0 ∩ L1 andwhose boundary operator ∂ is defined by ‘counting’ the number of solutions u :R× [0, 1] → M of the (nonlinear) Cauchy-Riemann equation

(1.3)

{∂u∂τ + J ∂u

∂t = 0

u(τ, 0) ∈ L0, u(τ, 1) ∈ L1.

The moduli space of pseudo-holomorphic strips entering in this counting problemis an appropriate compactification of the solution space.

To properly defined the Floer cohomology group, we need to study:

(1) (Filtration): a filtration of the Floer’s chain complex CF (L1, L0)(2) (Z-Grading): a Z-grading with respect to which ∂ has degree 1(3) (Sign): a sign on the generators which induces a Z-module (or at least

Q vector space) structure on CF (L1, L0) with respect to which ∂ is aZ-module (resp. a Q vector space) homomorphism.

In all of these structures, the relative version, i.e., the ‘difference’ between two gen-erators q, p ∈ L0∩L1 is canonically defined : for (1) it is the symplectic area, for (2)it is nothing but the so called Maslov-Viterbo index [V, Fl2] and for (3) it is basedon the choice of orientation of the determinant bundle detDu∂ → M(p, q;L0, L1)at a solution u of (1.3). More precisely, the gluing formula for the indices sharesa similar behavior with the problems on (1) and (2). (See Remarks 6.6 and 6.8.)Denote any of these invariants associated to the Floer trajectory u by I(q, p;u).The main problem to solve to provide these structures then is to see if there existssome family of functions I = I(q) independent of the choice of u ∈ M(p, q;L0, L1)such that

(1.4) I(q, p;u) = I(p)− I(q).

This is not possible in general unless one puts various restrictions on the triple(L0, L1;M): for (1) exactness of (M,ω) and of (L0, L1), for (2) vanishing of c1of (M,ω) and of the associated Maslov indices of L0, L1 and for (3) spinness ofthe pair L0, L1 or (more generally relative spinness of the pair (L0, L1)). Underthese restrictions respectively, it has been well understood by now that such achoice is always possible. See [Fl1], [Se2] for (1), [Fl2] and [FOOO00] for (2)and [FOOO00] for (3) respectively. (See also [Fu2] where (1), (2) and (3) aredescribed in the setting of Fukaya category. There are some technical errors and/orinconsistency with [FOOO09], in the description of [Fu2], which are corrected inthis paper.)

In this paper we define a filtered A∞ category on each symplectic manifold,which is an anchored version of Fukaya category. (See Theorem 8.14.) Its ob-jects are anchored Lagrangian submanifolds (L, γ) equipped with some extra data:(bounding cochain, spin structure and grading.) The morphism is an (anchored ver-sion of) Floer’s chain complex. We however emphasize that the cohomology groupHF ((L1, γ1), (L0, γ0)) is different from usual Floer cohomology group HF (L1, L0)which is defined in [FOOO00, FOOO09]. Namely the former is a component ofthe latter where only one of the connected components of Ω(L0, L1) is used for theconstruction. The anchored versions of (higher) compositions mk are also different

17


from the usual one. The precise relationship between the anchored version and thenon-anchored one is rather complicate to describe.

Necessity of studying this non-canonicality of filtration appears in several sit-uations: one is in the construction of Seidel’s long exact sequence as studied in[Oh3] and the other is in the study of Galois symmetry in Floer homology [Fu3].

Leaving the first problem to [Oh3], we will discuss the latter problem in Section10 of this paper. This involves the detailed discussion of the universal Novikov ring.An element of the universal Novikov ring has the form

(1.5)∑i

aieμiTλi

which is either a finite sum or an infinite sum with λi ≤ λi+1, limi→∞ λi = +∞.Here ai is an element of a ground ring R (for example R = Q) and λi are realnumbers. We consider the subring consisting of elements (1.5) such that λi ∈ Q inaddition. We denote it by Λrat

0,nov. We say that a Lagrangian submanifold L of asymplectic manifold M is rational if the subgroup

Γω(M,L) = {ω(α) | α ∈ π2(M,L)} ⊂ R

is discrete.Now we assume [ω] ∈ H2(M ;Q). Then there exists mamb ∈ Z+ and a com-

plex line bundle P with connection ∇ such that the curvature form of (P,∇) is2π

√−1mambω. We call (P,∇) the pre-quantum bundle. A Lagrangian submani-

fold L is called Bohr-Sommerfeld rational or simply BS-rational if holonomy groupof the restriction of (P,∇) to L is of finite order. (Such a Lagrangian submanifoldis called ‘cyclic’ in [Oh1] and just ‘rational’ in [Fu3]. Since a Lagrangian subman-ifold L is called a Bohr-Sommerfeld orbit when the holonomy group is trivial, thename ‘BS-rational’ seems to be a more reasonable choice.)

Theorem 1.2. To each (M,ω) with [ω] ∈ H2(M ;Q) with c1(M) = 0, wecan associate a filtered A∞ category with Λrat

nov coefficients. Its object consists ofa system (L,L, sp, b, s, SL) where L is a BS-rational Lagrangian submanifold of Mwith its Maslov class μL = 0, L is a flat U(1) bundle on L with finite holonomygroup, sp is a spin structure of L, b is a bounding cochain, s is a Z-grading, andSL is a rationalization of L. We denote this category by Fukrat(M,ω).

If mamb[ω] ∈ H2(M ;Z), then there exists a mambZ action on this category

which is compatible with the Z action of Λrat0,nov as continuous Galois group.

We will explain the notions appearing in the theorem in Section 10. In factour attempt to further reduce to a smaller ring leads us to considering a collectionof Lagrangian submanifolds for which one can associate an A∞ category over aNovikov ring like Q[[T 1/m]][T−1][e, e−1].

Theorem 1.3. Let (M,ω) be rational and (P,∇) be the pre-quantum line bun-dle of mambω. Then for each fixed N ∈ Z+, there exists a filtered A∞ categoryFukN (M,ω) over the ring C[[T 1/N ]][T−1][e, e−1]:

(1) its objects are (L,L, sp, b, SL) where L is a N BS-rational Lagrangian sub-manifold, L is a flat complex line bundle with its holonomy group G(L,∇)in {exp(2πk

√−1/N) | k ∈ Z}.

(2) The set of morphisms between two such objects is

CF (L1,L1, b1, sp1, SL1), (L0,L0, b0, sp0, SL0

);C[[T 1/N ]][T−1][e, e−1]).

18


There are also the anchored versions of Theorems 1.2, 1.3. See Subsetion 10.3.The category could be empty for some N . For example, one necessary condition

for FukN (M,ω) to be non-empty is that N should be divided by mamb. This leadsus to the notions of N-rational Lagrangian submanifolds: L is called N-rational if(P⊗N/mamb ,∇⊗N/mamb)|L is trivial. Then FukN (M,ω) is generated by N -rationalLagrangian submanifolds for each fixed N . The following question seems to beinteresting to study

Question 1.4. Is FukN (M,ω) generated by a finite number of objects? Morespecifically, is the number of the Hamiltonian isotopy class of compact BS N -rational Lagrangian submanifolds finite?

It is shown in Section 10 that the system (FukN (M,ω);<) with respect to thepartial order ‘N < N ′ if and only if N |N ′’ forms an inductive system. By definition,Fukrat(M,ω) will be the corresponding inductive limit.

The notion of anchored Lagrangian submanifolds has its origin in our preprint[FOOO00] Section 2.1 (see Remark 2.25 [FOOO00]) where the based point ofview of Lagrangian submanifolds is used in relation to construction of the Novikovcovering space and the action functional. (See also Remark 2.4.45 [FOOO09].)

The notion was first presented by the second named author in Eliashberg’s60th Birthday Conference: “New Challenges and Perspectives in Symplectic FieldTheory” held at Stanford University, June 25-29, 2007 and several other times invarious seminars and conferences afterwards in relation to construction of Seidel’slong exact sequence on closed Calabi-Yau manifolds [Oh3]. It was presented ina form somewhat different from the one used in this paper, though. During thepreparation of this article, Welschinger used the notion of based Lagrangian sub-manifolds in a recent preprint [W] with somewhat different framework and scopewhich has some overlap with the present paper (see Remark 5.5). However thenotion of based Lagrangian submanifolds itself is essentially the same as the onementioned in [FOOO00].

2. Novikov rings

The following ring was introduced in [FOOO00] which plays an important rolein the rigorous formulation of Lagrangian Floer theory.

Definition 2.1 (Universal Novikov ring). Let R be a commutative ring withunit. (In many cases, we take R = Q.) We define

Λnov =

{ ∞∑i=1

aiTλieμi/2

∣∣∣ ai ∈ R, λi ∈ R, λi ≤ λi+1, limi→∞

λi = ∞}

(2.1)

Λ0,nov =

{ ∞∑i=1

aiTλieμi/2 ∈ Λnov

∣∣∣ λi ≥ 0

}.(2.2)

There is a natural filtration on these rings provided by the multiplicative non-Archimedean valuation defined by

(2.3) v

( ∞∑i=1

aiTλieμi/2

):= inf {λi | ai �= 0} .

Here we assume (λi, μi) �= (λj , μj) for i �= j.

19


(2.3) is well-defined by the definition of the Novikov ring and induces a filtrationFλΛnov := v−1([λ,∞)) on Λnov. The function e−v : Λnov → R+ also provides anatural non-Archimedean norm on Λnov.

Let C be a free R module. We consider C ⊗R Λnov or C ⊗R Λ0,nov. We define

valuation v on it by v(∑

xiei) = inf v(xi), where ei is the basis of C. It definesa metric. We take the completion and denote it by C ⊗R Λnov or C ⊗R Λ0,nov,respectively.

In the point of view of Novikov’s Morse theory of closed one forms, it is naturalto use the version of Novikov ring that is a completion of the group ring of anappropriate quotient group of the fundamental group of Ω(L0, L1). This is the pointof view taken in many classical references of various Floer theories. The universalNovikov ring introduced above is slightly different from this Novikov ring. In thispaper we also use this more traditional Novikov ring, whose definition is now inorder.

We consider the space of paths (1.1), on which we are given the action one-formα (1.2). By definition

Zero(α) = {p : [0, 1] → M | p ∈ L0 ∩ L1, p ≡ p}.

Note that Ω(L0, L1) is not connected but has countably many connected compo-nents. We pick up a based path �01 ∈ Ω(L0, L1) and consider the correspond-ing component Ω(L0, L1; �01). We now review the definition of Novikov covering

we used in Section 2. Let g : Ω(L0, L1; �01) → Ω(L0, L1; �01) be an element of

deck transformation group of the universal cover Ω(L0, L1; �01) of Ω(L0, L1; �01).It induces a map w : [0, 1]2 → M with w(0, t) = �01(t) = w(1, t), w(s, 0) ∈ L0,w(s, 1) ∈ L1. (Namely s �→ w(s, ·) represent the path corresponding to g.) We put

(2.4) E(g) =

∫[0,1]2

w∗ω.

We also obtain a Lagrangian loop αλ01;λ01defined on ∂[0, 1]2 by

(2.5)αλ01;λ01

(0, t) = αλ01;λ01(1, t) = λ01(t),

αλ01;λ01(s, 0) = Tw(s,1)L0, αλ01;λ01

(s, 1) = Tw(s,1)L1.

Here λ01 is any path of Lagrangian subspaces along �01 with

λ01(0) = T�01(0)L0, λ01(1) = T�01(1)L1.

We denote by μ(g) be the Maslov index of this Lagrangian loop. See Section 5.1for the definition of Maslov index of this Lagrangian loop. We remark that thisindex does not depend on the choice of λ01 and can be expressed as the index of abundle pair over the annulus independently of this choice. (See Proposition 2.1.14[FOOO09].)

Definition 2.2. The Novikov covering is the covering space of Ω(L0, L1; �01)which corresponds to the kernel of the homomorphism

(E, μ) : π1(Ω(L0, L1; �01)) → R× Z.

Since Π(L0, L1; �01) is the deck transformation group of Novikov covering itfollows that there exists an (injective) group homomorphism

(2.6) (E, μ) : Π(L0, L1; �01) → R× Z.

20


Let Λnov be the field of fraction of Λ0,nov. (E, μ) induces a ring homomorphism

(2.7) Λ(L0, L1; �01) → Λnov

by ∑g

cg[g] �→∑

cgeμ(g)/2TE(g).

On Ω(L0, L1; �01) we have a unique single valued action functional A such that

dA = π∗α, A([�01]) = 0

where [�01] is a base point of Ω(L0, L1; �01).We then denote by Π(L0, L1; �01) the group of deck transformations. We de-

fine the associated Novikov ring Λ(L0, L1; �01) as a completion of the group ringQ[Π(L0, L1; �01)].

Definition 2.3. Λk(L0, L1; �01) denotes the set of all (infinite) sums∑

g∈Π(L0,L1;�01)

μ(g)=k

ag[g]

such that ag ∈ Q and that for each C ∈ R, the set

#{g ∈ Π(L0, L1; �01) | E(g) ≤ C, ag �= 0} < ∞.

We put Λ(L0, L1; �01) =⊕

k Λk(L0, L1; �01).

We call this graded ring the Novikov ring of the pair (L0, L1) relative to thepath �01. Note that this ring depends on the connected component of �01.

3. Anchors and abstract index

In this paper we always assume that L0 intersects L1 transversely.Let p, q ∈ L0∩L1. We denote by π2(p, q) = π2(p, q;L0, L1) the set of homotopy

classes of smooth maps u : [0, 1]× [0, 1] → M relative to the boundary

u(0, t) ≡ p, u(1, t) = q; u(s, 0) ∈ L0, u(s, 1) ∈ L1

and by [u] ∈ π2(p, q) the homotopy class of u and by B a general element inπ2(p, q). For given B ∈ π2(p, q), we denote by Map(p, q;B) the set of such w’s inclass B. Each element B ∈ π2(p, q) induces a map given by the obvious gluing map[p, w] �→ [q, w#u] for u ∈ Map(p, q;B). There is also the natural gluing map

(3.1) π2(p, q)× π2(q, r) → π2(p, r)

induced by the concatenation (u1, u2) �→ u1#u2. These ‘relative’ homotopy classesare canonically defined.

On the other hand, if we have chosen a base path �01 ∈ Ω(L0, L1), then we candefine the set of path homotopy classes of the maps w : [0, 1]2 → M satisfying theboundary condition

(3.2) w(0, t) = �01(t), w(1, t) ≡ p, w(s, 0) ∈ L0, w(s, 1) ∈ L1.

We denote the corresponding set of homotopy classes of the maps by π2(�01; p).Then we have the obvious gluing map

(3.3) π2(�01; p)× π2(p, q) → π2(�01; q); (α,B) �→ α#B.

21


Now we would like to generalize this construction for a chain L = (L0, · · · , Lk)of more than two Lagrangian submanifolds, i.e., with k ≥ 2. (We call such L theLagrangian chain and k + 1 the length of L.)

To realize this purpose, we use the notion of anchors of Lagrangian submani-folds in this paper.

Definition 3.1. Fix a base point y of ambient symplectic manifold (M,ω).Let L be a Lagrangian submanifold of (M,ω). We define an anchor of L to y is apath γ : [0, 1] → M such that γ(0) = y, γ(1) ∈ L. We call a pair (L, γ) an anchoredLagrangian submanifold.

A chain E = ((L0, γ0), · · · , (Lk, γk)) is called an anchored Lagrangian chain.L = (L0, · · · , Lk) is called its underlying Lagrangian chain.

It is easy to see that any homotopy class of path in Ω(L,L′) can be realizedby a path that passes through the given point y. Motivated by this observation,when we are given a Lagrangian chain (L0, L1, · · · , Lk) we also consider a chain ofanchors γi : [0, 1] → M of Li to y for i = 0, · · · , k. These anchors give a systematicchoice of a base path �ij ∈ Ω(Li, Lj) by concatenating γi and γj :

(3.4) �ij(t) =

{γi(1− 2t) t ≤ 1/2

γj(2t− 1) t ≥ 1/2.

The upshot of this construction is the following overlapping property

(3.5) �ij(t) = �i�(t) for 0 ≤ t ≤ 1

2, �ij(t) = ��j(t) for

1

2≤ t ≤ 1

for all j, �.Let (L0, · · · , Lk) be a Lagrangian chain and p(i+1)i ∈ Li ∩ Li+1. (p(k+1)k =

p0k and Lk+1 = L0 as convention.) We write �p = (p10, · · · , pk(k−1)). Let χi =

exp(−2πi√−1/k). We consider the set of homotopy class of maps v : D2 → M

such that v(χi+1χi) ⊂ Li and v(χi) = pi(i+1). We denote it by π2(L; �p). If E isan anchored Lagrangian chain and L be its underlying Lagrangian chain we writeπ2(E ; �p) in place of π2(L; �p) some times by abuse of notation.

Definition 3.2. Let E = {(Li, γi)}0≤i≤k be a chain of anchored Lagrangiansubmanifolds. A homotopy class B ∈ π2(E ; �p) is called admissible to E if it can beobtained by a polygon that is a gluing of k bounding strips w−

i(i+1) : [0, 1]× [0, 1] →M satisfying

w−i(i+1)(s, 0) ∈ Li, w−

i(i+1)(s, 1) ∈ Li+1(3.6a)

w−i(i+1)(0, t) = p(i+1)i.(3.6b)

w−i(i+1)(1, t) =

{γi(1− 2t) 0 ≤ t ≤ 1

2

γi+1(2t− 1) 12 ≤ t ≤ 1.

(3.6c)

When this is the case, we denote the homotopy class B as

B = [w−01]#[w−

12]# · · ·#[w−k0]

and the set of admissible homotopy classes by πad2 (E ; �p).

We note that not all homotopy classes in π2(E ; �p) is admissible for a givenanchored Lagrangian chain. (See however Lemma 3.6.)

22


Definition 3.3. Let (Li, γi), i = 0, 1 be anchored Lagrangian submanifolds.We say p ∈ L0 ∩ L1 is admissible (with respect to the pair ((L0, γ0), (L1, γ1))) ifthere exists w = w01 satisfying (3.6a) for i = 0 and (3.6b) for i = 0, p10 = p.

Note p is admissible if and only if π2(�01; p) is nonempty. (Here �01 is as in(3.4).) Let us go back to the case k = 1. First note that we have:

(3.7) π2(p, q) = π2(L; �p)

where L = (L0, L1), �p = (p, q) and the left hand side is as in the beginning of thissection.

Lemma 3.4. Let k = 1 and E = ((L0, γ0), (L1, γ1)). Then πad2 (E , (p, q)) =

π2(E , (p, q)) if p, q are admissible. Otherwise πad2 (E , (p, q)) is empty.

The proof is easy and omitted.

Lemma 3.5. Let L0, L1 be a pair of Lagrangian submanifold and p ∈ L0 ∩ L1.Then for each given anchor γ0 of L0 there exists an anchor γ1 of L1 such that p isadmissible with respect to the pair ((L0, γ0), (L1, γ1)).

The proof is easy and omitted.The proof of the following two lemmas are also easy and so is omitted.

Lemma 3.6. Let L be a Lagrangian chain and B ∈ π2(L; �p). Then there existanchors γi of Li (i = 0, · · · , k) such that B is admissible with respect to E , where

E = ((L0, γ0), · · · , (Lk, γk)).

The anchors in Lemmas 3.5, 3.6 are not necessarily unique (even up to ho-motopy). It is rather complicated to describe how many there are. (See Section9 for some illustration.) The following definition can be used to study the gluingformulas of symplectic areas and Maslov indices of pseudo-holomorphic polygonsthat enter in the construction of the anchored version of Fukaya category.

Definition 3.7. Let R be a module. We say a collection of maps

I = {Ik : πad2 (E ; �p) → R}∞k=1

an abstract index over the collection of anchored Lagrangian chains E , if they satisfythe following gluing rule: whenever the gluing is defined, we have

Ik+1([w−01]# · · ·#[w−

(k−1)k]#[w−k0]) =

k∑i=0

I1([w−i(i+1)]).

In subsection 10.3, we will use another abstract index, a normalized symplecticarea over the class of BS-rational Lagrangian submanifolds with R = Q or withR = 1

N · Z for integers N .

4. Anchors, action functional and action spectrum

For given two anchors γ, γ′ homotopic to each other, we denote by π2(γ, γ′;L)

the set of homotopy classes of the maps w : [0, 1]2 → M satisfying

w(0, t) = γ(t), w(1, t) = γ′(t), w(s, 0) ≡ y, and w(s, 1) ∈ L.

For any such map w, we define

(4.1) a(γ,γ′;L)(w) =

∫w∗ω.

23


It is immediate to check that this function pushes down to π2(γ, γ′ : L) which we

again denote by a(γ,γ′;L).We denote by G(γ, γ′ : L) ⊂ R the image of a(γ,γ′;L). The following is easy

to check whose proof we omit. Let γ ∗ γ′ be an element Ω(L,L;M)) obtained byconcatenating γ (where γ(t) = γ(1 − t)) and γ′ in the same way as (3.4), andΩγ∗γ′(L,L;M)) the connected component of Ω(L,L;M)) containing it.

Lemma 4.1. π2(γ, γ′ : L) is a principal homogeneous space of π1(Ωγ∗γ′(L,L;M)).

and so G(γ, γ′ : L) is a principal homogeneous space of the group

{ω(C) | C ∈ π1(Ωγ∗γ′(L,L;M))}.

The action functional A = A(γ0,γ1;L) : Ω(L0, L1; �01) → R is defined by

A([�, w]) =

∫w∗w.

Note an element of Ω(L0, L1; �01) is identified with a pair [�, w] where � ∈Ω(L0, L1; �01) and w : [0, 1]2 → M satisfies

(4.2) w(0, t) = �01(t), w(1, t) ≡ �(t), w(s, 0) ∈ L0, w(s, 1) ∈ L1.

We identify [�, w] with [�, w′] if

(4.3)

∫(w′#w)∗ω = 0, μ(w′#w) = 0.

Here w′(s, t) = w′(1 − s, t) and μ is an appropriate Maslov index. (See (2.5) andDefinition 2.2.)

We now study dependence of the action functional A(L0,γ0),(L1,γ1) on theiranchors. Let γ0, γ

′0 and γ1, γ

′1 be two anchors of L0 and L1 respectively. It defines

�01 and �′01 by (3.4). We assume that there exist paths w0, w1 connecting themrespectively. Then w0#w1 induces a diffeomorphism

Φw0#w1: Ω(L0, L1; �01) → Ω(L0, L1; �

′01)

defined by

(4.4) Φw0#w1([�, u]) = [�, (w0#w1)#u].

For the clarity of notations, we will use # for two dimensional concatenations andby ∗ for one dimensional ones.

Proposition 4.2. Let γi, γ′i and wi for i = 0, 1 be as above. Consider [�, u] ∈

Ω(L0, L1; �01). Then we have

A(L0,γ0),(L1,γ1) − Φ∗w0#w1

A(L0,γ′0),(L1,γ′

1)≡ ω([w0#w1]).

Proof. Obvious from the definition. �

Now we define:

Definition 4.3 (Action spectrum). Denote by Spec((L0, γ0), (L1, γ1)) the setof critical values of A(L0,γ0),(L1,γ1) and call the action spectrum of the pair (L0, γ0),(L1, γ1).

An immediate corollary of Proposition 4.2 and this definition is the following

24


Corollary 4.4. We assume that γi is homotopic to γ′i for i = 0, 1. Then

there exists a real constant c = c((L0, γ0), (L1, γ1); (L0, γ′0), (L1, γ

′1)) depending on

the pair (L0, γ0), (L1, γ1) such that

Spec((L0, γ0), (L1, γ1)) = Spec((L0, γ′0), (L1, γ

′1)) + c

as a subset of R.

Proof. Let γi, γ′i and wi for i = 0, 1 be as above. By Proposition 4.2, we

haveCritA(L0,γ′

0),(L1,γ′1)+ ω([w0#w1]) = CritA(L0,γ0),(L1,γ1)

for any choice of w0, w1 joining γ0, γ′0 and γ1, γ

′1 respectively.

Just take c = ω([w0#w1]). This finishes the proof. �

Next we consider the Lagrangian chains with 3 or more elements in them. Whenwe are given an anchored Lagrangian chain

E = ((L0, γ0), (L1, γ1), · · · , (Lk, γk))

these anchors give a systematic choice of a base path �ij ∈ Ω(Li, Lj) by concatenat-ing γi and γj as in (3.4). Inside the collection of anchored Lagrangian submanifolds(L, γ) we are given a coherent system of single valued action functionals

A : Ω0(Li, Lj ; �ij) → R.

We will use the action functional associated to �ij to define an energy level on thecritical point set A : CritA → R. By the overlapping property (3.5), the followingproposition is immediate whose proof we omit.

Proposition 4.5. Denote by E an anchored Lagrangian chain. Consider themap Iω,k : π2(�p; E) → R defined by the symplectic area Iω,k(α) = ω(α) for k =1, · · · ,. Then the collection denoted by Iω = {Iω,k}∞k=1 defines an abstract index ofanchored Lagrangian chains.

5. Grading and filtration

A familiar description of generators of Floer chain module as the set of equiv-alence classes [p, w] in the Novikov covering space is useful as far as the study offiltration on the Floer complex is concerned. However for the study of gradingand signs on the Floer complex, we have to have additional structures on the Floerchain module which requires some geometric condition on the Lagrangian side, e.g.,spin structure or graded structure. There has been a few different approach to howone incorporates these additional structures. In this section, we describe them byusing anchors.

5.1. Maslov index in Lagrangian Grassmannian. In this subsection, wereview the definition of Maslov index in Lagrangian Grassmannian. The LagrangianGrassmannian Lag(S, ω) of a symplectic vector space (S, ω) is defined to be

Lag(S, ω) = {V | V is a Lagrangian subspace of (S, ω)}.When we equip S a compatible complex structure J and define U(S) to be the groupof unitary transformations of S, any V0, V1 ⊂ Lag(S, ω) can be written as V1 = A·V0

for some A ∈ U(S). In [A], this fact is used to show that H1(Lag(S, ω),Z) ∼= Z.It generator μ ∈ H1(Lag(S, ω),Z) is the Maslov class [A] and two loops γ1, γ2 arehomotopic if and only if μ(γ1) = μ(γ2).

25


We give an elementary description of the Maslov class below. We fix V0 ∈Lag(S, ω) and put

Lag1(S, ω) = {V ∈ Lag(S, ω) | dim(V ∩ V0) ≥ 1}.It is proven in [A] that Lag1(S, ω) is co-oriented and so defines a cycle whosePoincare dual is precisely the Maslov class μ ∈ H1(Lag1(S, ω),Z).

The tangent space TV0Lag(S, ω) is canonically isomorphic to the set of qua-

dratic forms on V0.

Definition 5.1. We say any tangent vector pointing the chamber of nonde-generate positive-definite quadratic forms is positively directed.

The following is also proved in [A].

Lemma 5.2. There exists a neighborhood U of V0 ∈ Lag(S, ω), the set

U \ Lag1(S, ω;V0)

has exactly n+ 1 connected components each of which contains V0 in its closure.

We refer readers to [A] or see Proposition 2.1.3 of [FOOO09] for the proof ofthe following proposition.

Proposition 5.3. Let (S, ω) be a symplectic vector space and V0 ∈ (S, ω) be agiven Lagrangian subspace. Let V1 ∈ Lag(S, ω)\Lag1(S, ω;V0) i.e., be a Lagrangiansubspace with V0∩V1 = {0}. Consider smooth paths α : [0, 1] → Lag(S, ω) satisfying

(1) α(0) = V0, α(1) = V1.(2) α(t) ∈ Lag(S, ω) \ Lag1(S, ω;V0) for all 0 < t ≤ 1.(3) α′(0) is positively directed.

Then any two such paths α1, α2 are homotopic to each other via a homotopy s ∈[0, 1] �→ αs such that each αs also satisfies the 3 conditions above.

Let Lag+(S, ω) be the double cover of Lag(S, ω). Its element is regarded as anelement V of Lag(S, ω) equipped with an orientation of V .

5.2. Anchors and grading. To use the anchor in the definition of a gradingin the Floer complex, we need to equip each anchor with an additional decoration.

Let y ∈ M be the base point. We fix an oriented Lagrangian subspace Vy ∈Lag+(TyM).

Definition 5.4. Consider an anchored Lagrangian (L, γ). We denote by λ asection of γ∗Lag+(M,ω) such that

λ(0) = Vy, λ(1) = Tγ(1)L.

We call such a pair (γ, λ) a graded anchor of L (relative to (y, Vy)) and a triple(L, γ, λ) a graded anchored Lagrangian submanifold.

Remark 5.5. We remark that a notion similar to the graded anchor also ap-pears in Welschinger’s recent work [W].

Let (L0, γ0, λ0) and (L1, γ1, λ1) be graded anchored Lagrangian submanifoldsrelative to (y, Vy). Assume that L0 and L1 intersect transversely. We define λ01(t) ∈Lag+(T�01(t)M) by concatenating λ0 and λ1 as follows:

(5.1) λ01(t) =

{λ0(1− 2t) t ≤ 1/2

λ1(2t− 1) t ≥ 1/2.

26


We consider a pair [p, w] where p ∈ L0 ∩ L1, and w : [0, 1]2 → M as in (3.2). Toput a grading at [p, w], we recall the definition of Maslov-Morse index introduced in[FOOO00] (see [FOOO09] Section 2.2.2). For given w, we associate a Lagrangianloop α[p,w];λ01

defined on ∂[0, 1]2 by

(5.2)α[p,w];λ01

(0, t) = λ01(t), α[p,w];λ01(s, 0) ≡ Tw(s,0)L0,

α[p,w];λ01(s, 1) ≡ Tw(s,1)L1, α[p,w];λ01

(1, t) = α+p (t)

where α+p : [0, 1] → TpM is a path connecting from TpL0 to TpL1 in Lag(TpM,ωp)

whose homotopy class is the unique one as described in Proposition 5.3.Let p ∈ L0 ∩ L1 and w : [0, 1]2 → M satisfy (3.2). Choose a symplectic

trivialization Φ = (π, φ) : w∗TM → [0, 1]2 × TpM ∼= [0, 1]2 × R2n where π :

w∗TM → [0, 1]2 and φ : w∗TM → TpM are the corresponding projections to [0, 1]2

and TpM respectively. Φ is homotopically unique. Now we denote by αΦ[p,w];λ01

the

Lagrangian loop

αΦ[p,w];λ01

= φ(α[p,w];λ01◦ c).

Here we fix a piecewise smooth parametrization c : S1 ∼= R/Z → ∂[0, 1]2 of ∂[0, 1]2

with positive orientation with c(0) = (1, 0).

Definition 5.6. We define the Maslov-Morse index, denoted by μ([p, w];λ01),to be the Maslov index of this Lagrangian loop αΦ

[p,w];λ01in (TpM,ω).

This definition does not depend on the trivialization Φ or on the (positive)parametrization c of ∂[0, 1]2 and so well-defined.

Remark 5.7. Here and hereafter we uses the symbol λ for a path in Lag+,the oriented Lagrangian Grassmannian and α for a path in Lag, the un-orientedLagrangian Grassmannian.

Lemma 5.8. Let p, w, λ01 be as in Definition 5.6. We put

w−(s, t) = w(s, 1− t), λ10(t) = λ01(1− t).

Then

(5.3) μ([p, w];λ01) + μ([p, w−];λ10) = n.

Proof. Let Φ and αΦ[0,1];λ01

be as above. If we denote ι : [0, 1]2 → [0, 1]2 to

be the map ι(s, t) = (s, 1 − t), we have w− = w ◦ ι. Therefore we can trivialize(w−)∗TM = ι∗w∗TM by the map Φ− : (w−)∗TM → [0, 1]2 × R2n defined byΦ− = Φ ◦ ι∗. Then

˜αΦ[p,w];λ01

= φ(α[p,w];λ01◦ c)

where (·) denotes the inverse path, e.g., c(θ) = c(−θ). By definition of αΦ−

[p,w−];λ10,

the path φ(α[p,w];λ01◦ c) coincides with αΦ−

[p,w−];λ10(up to parametrization) except

on the segment c−1({1} × [0, 1]).

Therefore the composition αΦ[p,w];λ01

∗ αΦ−

[p,w−];λ10of αΦ−

[p,w−];λ10and αΦ

[p,w];λ01

is homotopic to a path α = α− ∪ α+ : S1 ∼= I− ∪ I+ → Lag(R2n, ω0) with I± ={1}× [0, 1] such that both α± : I± → Lag(R2n, ω0) are the paths positively directedat t = 0 provided in Proposition 5.3 and satisfy

α+(0) = α−(1) = φ(TpL0), α+(1) = α−(0) = φ(TpL1).

27


It is easy to see that the Maslov index of such α is n and hence we obtain

(5.4) n = μ(αΦ[p,w];λ01

∗ αΦ−

[p,w−];λ10) = μ(αΦ

[p,w];λ01) + μ(αΦ−

[p,w−];λ10).

By definition, the last sum is nothing but μ([p, w];λ01) + μ([p, w−];λ10). Thisfinishes the proof of (5.3). �

5.3. Polygonal Maslov index. Consider a chain of Lagrangian submanifoldsL = (L0, · · · , Lk) and a chain of intersection points (p0k, pk(k−1), · · · , p10) withpi(i−1) ∈ Li−1 ∩ Li for i = 0, · · · , k. We consider the disc with marked points

{z0k, zk(k−1), · · · , z10} and denote D2 = D2 \ {z0k, zk(k−1), · · · , z10}. We assume

z0k, zk(k−1), · · · , z10 respects counter clock-wise cyclic order of ∂D2.

Remark 5.9. Here and hereafter the suffix j is regarded as modulo k + 1.Namely p(j+1)j in case j = k means p0k, for example. We also put pij = pji.

For the following discussion, we will consider the cases k ≥ 1, i.e, the caseswith lengthL ≥ 2.

For each given such chains, we define the set of maps

C∞(D2, E ; �p), �p = {p0k, pk(k−1), · · · , p10}

to be the set of all w : D2 → M such that

(5.5) w(z(j+1)jzj(j−1)) ⊂ Lj , w(zj(j−1)) = pj(j−1) ∈ Lj ∩ Lj−1,

and that it is continuous on D2 and smooth on D2. We will define a topologicalindex, which is associated to each homotopy class B ∈ π2(L; �p). We denote it by

μ(L, �p;B). Let w ∈ C∞(D2,L; �p) be a map such that [w] = B. We denote by

F(L; �p;B) ⊂ C∞(D2,L; �p)

the set of such maps.

We identify [0, 2π]/(0 ∼ 2π) ∼= S1 by t �→ e√−1t. (The direction t increase then

becomes counter-clockwise order of S1.)Under a symplectic trivialization of the bundle w∗TM , the map

αw : S1 = ∂D2 → Lag(R2n, ω0); t �→ Tw(t)Li if t ∈ z(i+1)izi(i−1)

defines a piecewise smooth path with discontinuities at (k+1) points zi(i−1) ∈ ∂D2

for i = 0, 1, · · · , k, at which we have

(5.6) limt→zi(i−1)−0

αw(t) = Tpi(i−1)Li, lim

t→zi(i+1)+0αw(t) = Tpi(i−1)

Li−1

By the transversality hypothesis, Tpi(i−1)Li and Tpi(i−1)

Li−1 are Lagrangian

subspaces in (R2n, ω0) with Tpi(i−1)Li ∩ Tpi(i−1)

Li−1 = {0}. We fix a smooth path

α−i(i−1) : [0, 1] → Lag(R2n, ω0) for each i = 0, · · · , k so that

(5.7) α−i(i−1)(0) = Tpi(i−1)

Li, α−i(i−1)(1) = Tpi(i−1)

Li−1

and −(α−i(i−1))

′(1) is positively directed in the sense of Definition 5.1.

In other words α−i(i−1)(t) = α+

p(i−1)i(1 − t), where the right hand side is as in

(5.2). By Proposition 5.3, such a choice is unique up to homotopy relative to theend points t = 0, 1. Inserting α−

i(i−1) into the map αw at each zi(i−1), we obtain a

continuous loop αw in Lag(R2n, ω0).

28


Definition 5.10. Let L = (L0, · · · , Lk) be a Lagrangian chain. We define thetopological index, denoted by μ(L, �p;B), to be the Maslov index of the loop αw,i.e.,

μ(L, �p;B) = μ(αw).

This definition is essentially reduced to the Maslov-Viterbo index [V] for thepairs (L0, L1) when k = 1 and reduces to the one given in Section A3 [KS] for thecase where Li are all affine.

Remark 5.11. We remark that L0, L1, . . . , Lk are put on the boundary of thedisc D2 in a clockwise order. On the other hand, z0k, zk(k−1), . . . , z21, z10 are in thecounter clockwise order.

This is consistent with the case k = 1 discussed in [FOOO09]. (See Remark3.7.23 (1) [FOOO09].)

Now consider a chain of graded anchored Lagrangian submanifolds 0 k

Li = (Li, γi, λi). It induces a grading λij along �ij as in (5.1). We remark that λij

also satisfy the overlapping property

(5.8) λij |[0, 12 ] = λi�|[0, 12 ] λij |[ 12 ,1] = λ�j |[ 12 ,1].Let p(i+1)i = pi(i+1) ∈ Li ∩ Li+1. We put

(5.9) w+(i+1)i(s, t) = w−

i(i+1)(1− s, t)

where the right hand side is as in Definition 3.2.

Lemma 5.12. Let E be a graded anchored Lagrangian chain. Suppose B ∈πad2 (E , �p) given as Lemma 3.2. Then we have

(5.10) μ(L, �p;B) +

k∑i=0

μ([p(i+1)i, w+(i+1)i];λi(i+1)) = 0.

Proof. Since μ([p(i+1)i, w+(i+1)i];λi(i+1)) is defined as the Maslov index of the

loop α[p(i+1)i,w+(i+1)i

];λi(i+1)(Definition 5.6), the equality (5.10) follows from

k∑i=0

α[p(i+1)i,w+(i+1)i

];λi(i+1)+ αw ∼ 0,

where αw is as in Definition 5.10 with B = [w−01]# · · ·#[w−

(k−1)k]#[w−k0] and ∼

means homologous. �

When the length of E is k+1, we define μk(B) = μ(E , �v;B) where B ∈ πad2 (E ; �p).

Corollary 5.13. Define μ1 : π2(�01, p) → Z by setting μ1(α) := −μ([p, w];λ01)for a representative [p, w] of the class α ∈ π2(�01; p). Then the sequence of mapsμ = {μk}∞k=1 with μk : πad

2 (E ; �p) → Z, k ≥ 1 defines an abstract index.

6. Orientation

To be able to define various operators in Floer theory, we need to provide acompatible system of orientations on the Floer moduli spaces and other modulispaces of pseudo-holomorphic polygons. We here explain the way how we giveorientations on the moduli spaces, which is basically the same as that of Chapter8 [FOOO09].

29

E=(L ,· · · ,L ),


Definition 6.1. A submanifold L ⊂ M is called relatively spin if it is orientableand there exists a class st ∈ H2(M,Z2) such that st|L = w2(TL) for the Stiefel-Whitney class w2(TL) of TL.

A chain (L0, L1, · · · , Lk) or a pair (L0, L1) of Lagrangian submanifolds is saidto be relatively spin if there exists a class st ∈ H2(M,Z2) satisfying st|Li

= w2(TLi)for each i = 0, 1, · · · , k.

We fix such a class st ∈ H2(M,Z2) and a triangulation of M . Denote by M (k)

its k-skeleton. There exists a real vector bundle V (st) on M (3) with w1(V (st)) =0, w2(V (st)) = st. Now suppose that L is relatively spin and L(2) be the 2-skeletonof L. Then V ⊕ TL is trivial on the 2-skeleton of L. We define

Definition 6.2. We define a (M, st)-relative spin structure of L to be a choiceof V and a spin structure of the restriction of the vector bundle V ⊕ TL to L(2).

The relative spin structure of a chain of Lagrangian submanifolds (L0, · · · , Lk)is defined in the same way by using the same V for all Li.

Let p, q ∈ L0 ∩L1 and B ∈ π2(p, q). We consider u : R× [0, 1] → M such that

du

dτ+ J

du

dt= 0(6.1a)

u(τ, 0) ∈ L0, u(τ, 1) ∈ L1,

∫u∗ω < ∞(6.1b)

u(−∞, ·) ≡ p, u(∞, ·) ≡ q.(6.1c)

It induces a continuous map u : [0, 1]2 → M with u(0, t) ≡ p, u(1, t) ≡ q in an obvi-ous way. With an abuse of notation, we denote by [u] the homotopy class of the map

u in π2(p, q). We denote by M◦(p, q;B) the moduli space consisting of the maps u

satisfying (6.1) and compactify M◦(p, q;B)/R its quotient by the τ -translations byusing an appropriate notion of stable maps as in Section 3 [FOOO00] (see Section2.1 [FOOO09]). We denote the compactification by M(p, q;B). We call this theFloer moduli space. It carries the structure of a space with Kuranishi structure[FOn], Appendix A.1 [FOOO09].

If (L0, L1) is a relatively spin pair, then M(p, q;B) is orientable. Furthermorea choice of relative spin structures gives rise to a compatible system of orientationsfor M(p, q;B) for all pair p, q ∈ L0 ∩L1 and B ∈ π2(p, q). For completeness’ sake,we now recall from Section 8.1 [FOOO09] how the relative spin structure gives riseto a system of coherent orientations.

Let p ∈ L0 ∩ L1 and w satisfies (3.2) . We denote by Map(�01; p;L0, L1;α)the set of such maps [0, 1]2 → M its homotopy class [w] = α in π2(�01; p). Letw ∈ Map(�01; p;L0, L1;α). Let Φ : w∗TM → [0, 1]2 × TpM be a (homotopicallyunique) symplectic trivialization as before. The trivialization Φ, together with theboundary condition, w(0, t) = �01(t) and the Lagrangian path λ01 along �01, definesa Lagrangian path

λΦ = λΦ([p,w];λ01)

: [0, 1] → TpM

satisfying λΦ(0) = TpL0, λΦ(1) = TpL1. The homotopy class of this path does not

depend on the trivialization Φ but depends only on [p, w] and the homotopy classof λ01. Hereafter we omit Φ from notation.

We remark that relative spin structure determines a trivialization of Vλ01(0) ⊕Tλ01(0)L0 = Vλ01(0) ⊕ λ01(0) and Vλ01(1) ⊕TpL1 = Vλ01(1) ⊕ λ01(1). We take and fixaway to extend this trivialization to the family �∗01V ⊕ λ01 on [0, 1].

30


We consider the following boundary valued problem for the section ξ of w∗TMon R≥0 × [0, 1] of W 1,p class such that:

Dw∂(ξ) = 0(6.2a)

ξ(0, t) ∈ λ01(t), ξ(τ, 0) ∈ TpL0, ξ(τ, 1) ∈ TpL1.(6.2b)

Here Dw∂ is the linearization operator of the Cauchy-Riemann equation.We define W 1,p(R≥0 × [0, 1], TpM ;λ01) to be the set of sections ξ of w∗TM on

R≥0 × [0, 1] of W 1,p class satisfying (6.2b). Then (6.2a) induces an operator

Dw∂ : W 1,p(R≥0 × [0, 1], TpM ;λ) → Lp(R≥0 × [0, 1], TpM ⊗ Λ0,1),

which we denote by ∂([p,w];λ01). The following proposition was proved in Lemma3.7.69 [FOOO09].

Proposition 6.3. We have

(6.3) Index ∂([p,w];λ01) = μ([p, w];λ01).

We denote its determinant line by

det ∂([p,w];λ01).

By varying w in its homotopy class α ∈ π2(�01; p) = π2(�01; p;L0, L1), these linesdefine a line bundle

(6.4) det ∂([p,w];λ01) → Map(�01; p;L0, L1;α).

The bundle (6.4) is trivial if (L0, L1) is a relatively spin pair. (See Section 8.1[FOOO09].)

We need to find a systematic way to orient (6.4) for various α ∈ π2(�01; p)simultaneously. Following Subsection 8.1.3 [FOOO09] we proceed as follows. Letλp : [0, 1] → TpM be a path connecting from TpL0 to TpL1 in Lag+(TpM,ω). Therelative spin structure determines a trivialization of Vp ⊕ TpL0 = Vp ⊕ λp(0) andof Vp ⊕ TpL1 = Vp ⊕ λp(1). We fix an extension of this trivialization of the [0, 1]parametrized family of vector spaces Vp ⊕ λp. We define

(6.5) Z+ = {(τ, t) ∈ R2 | τ ≤ 0, 0 ≤ t ≤ 1} ∪ {(τ, t) | τ2 + (t− 1/2)2 ≤ 1/4}

Remark 6.4. We would like to remark that attaching the semi-disc to the sideof the semi-strip t = 0 is not necessary for the definition of Z+. However for theconsistency with the notation of Subsection 8.1.3 [FOOO09], we keep using Z+

instead of the simpler (−∞, 0]× [0, 1].

We consider maps ξ : Z+ → TpM of W 1,p class and study the linear differentialequation

∂ξ = 0(6.6a)

ξ(eπi(t−1/2)/2 + i/2) ∈ λp(t), ξ(τ, 0) ∈ TpL0, ξ(τ, 1) ∈ TpL1.(6.6b)

It defines an operator

W 1,p(Z+, TpM ;λp) → Lp(Z+;TpM ⊗ Λ0,1),

which we denote by ∂λp. Let Index ∂λp

be its index, which is a virtual vector space.The following theorem is proved in the same way as in Chapter 8 [FOOO09].

31


Theorem 6.5. Let (L0, L1) be a relatively spin pair of oriented Lagrangiansubmanifolds. Then for each fixed α the bundle (6.4) is trivial.

If we fix a choice of system of orientations op on Index ∂λpfor each p, then it

determines orientations on (6.4), which we denote by o[p,w].Moreover op, o[p,w] determine an orientation of M(p, q;B) denoted by o(p, q;B)

by the gluing rule

(6.7) o[q,w#B] = o[p,w]#o(p, q;B)

for all p, q ∈ L0 ∩ L1 and B ∈ π2(p, q) so that they satisfy the gluing formulae

∂o(p, r;B) = o(p, q;B1)#o(q, r;B2)

whenever the virtual dimension of M(p, r;B) is 1. Here ∂o(p, r;B) is the in-duced boundary orientation of the boundary ∂M(p, r;B) and B = B1#B2 andM(p, q;B1)#M(q, r;B2) appears as a component of the boundary ∂M(p, r;B).

Remark 6.6. In the last statement in Theorem 6.5, we assumed thatM(p, r;B)is one-dimensional. In general, we have

∂o(p, r;B) = (−1)εo(p, q;B1)#o(q, r;B2),

where ε = dimM(q, r;B2), which is presented in the proof of Proposition 8.7.3 in[FOOO09]. For the definition of the orientation of the moduli spaces for the filteredbimodule structure, see Sections 8.7 and 8.8 (Definition 8.8.11) in [FOOO09].

Proof. The first paragraph follows from Section 8.1 [FOOO09]. We glue theend (−∞, t) of Z+ with (+∞, t) of R≥0×[0, 1] to obtain (R≥0×[0, 1])#Z+. We ‘glue’

operators (6.2) and (6.6) in an obvious way to obtain an operatorDw∂(R≥0×[0,1])#Z+

on (R≥0× [0, 1])#Z+. We have an isomorphism of (family of) virtual vector spaces:

(6.8) Index (Dw∂(R≥0×[0,1])#Z+) ∼= Index ∂([p,w];λ01) ⊕ Index ∂λp

We fixed a trivialization of the family of vector spaces Vp ⊕ λp and �∗01V ⊕ λ01,which extends a trivialization of V ⊕ TL0, V ⊕ TL1 on the two skeletons of L0

and L1 respectively, which is given by the relative spin structure. It induces acanonical orientation of the index bundle Index (Dw∂(R≥0×[0,1])#Z+

) by Lemma

3.7.69 [FOOO09]. Therefore the orientation of Index ∂λpinduces an orientation of

Index ∂([p,w];λ01) in a canonical way. This implies the second paragraph.The third paragraph is a consequence of (6.7) which is similar to the proof of

Theorem 8.1.14 [FOOO09]. �

One can generalize the above discussion to the moduli space of pseudo-holomorphicpolygons in a straightforward way, which we describe below.

Consider a disc D2 with k + 1 marked points z0k, zk(k−1), · · · , z10 ⊂ ∂D2 re-

specting the counter clockwise cyclic order of ∂D2. We take a neighborhood Ui ofzi(i−1) and a conformal diffeomorphism ϕi : Ui \ {zi(i−1)} ⊂ D2 ∼= (−∞, 0] × [0, 1]of each zi(i−1). For any smooth map

w : D2 → M ;w(zi(i−1)) = pi(i−1), w(z(i+1)izi(i−1)) ⊂ Li

we deform w so that it becomes constant on ϕ−1i ((−∞,−1] × [0, 1]) ⊂ Ui, i.e.,

w(z) ≡ pi(i−1) for all z ∈ ϕ−1((−∞,−1]× [0, 1]). So assume this holds for w from

32


now on. We now consider the Cauchy-Riemann equation

Dw∂(ξ) = 0(6.9a)

ξ(θ) ∈ Tw(t)Li for θ ∈ z(i+1)izi(i−1) ⊂ ∂D2.(6.9b)

We remark that on Ui = (−∞, 0]× [0, 1] the boundary condition (6.9b) becomes

(6.10) ξ(s, 0) ∈ Li−1, ξ(s, 1) ∈ Li.

(6.9) induces a Fredholm operator, which we denote by

(6.11) ∂w;L : W 1,p(D2;w∗TM ;L) → Lp(D2;w∗ ⊗ Λ0,1).

Moving w we obtain a family of Fredholm operator ∂(L;p;B) parametrized by asuitable completion of F(�p;L;B) for B ∈ π2(�p;L). Therefore we have a well-defineddeterminant line bundle

(6.12) det ∂(L;p;B) → F(L; �p;B).

The following theorem is an extension of the above Theorem 6.5.

Theorem 6.7. Suppose L = (L0, · · · , Lk) is a relatively spin Lagrangian chain.Then each det ∂(p;L;B) is trivial.

Moreover we have the following: If we fix orientations opijon Index ∂λpij

as

in Theorem 6.5 for all pij ∈ Li ∩ Lj, with Li transversal to Lj, then we have asystem of orientations, denoted by ok+1(�p;L;B), on the bundles (6.12) so that it iscompatible with gluing map in an obvious sense.

Proof. Let w+(i+1)i ∈ π2(�i(i+1); p(i+1)i) be as in (5.9) and we consider the

operator ∂([p(i+1)i,w+(i+1)i

];λi(i+1)). We glue it with ∂(L;p;B) at Ui+1. ((6.10) implies

that the boundary condition can be glued.) After gluing all of ∂([p(i+1)i,w+(i+1)i

];λi(i+1))

we have an index bundle of a Fredholm operator

(6.13) ∂(L;p;B)#0∑

i=k

∂([p(i+1)i,w+(i+1)i

];λi(i+1)).

By adopting the argument used in section 8.1.3 [FOOO09], we show that the indexbundle of (6.13) has canonical orientation. On the other hand, the index virtualvector spaces of ∂([p(i+1)i,w

+(i+1)i

];λi(i+1))are oriented by Theorem 6.5. Theorem 6.7

follows. �

We can prove that the orientation of ∂(L;p;B) depends on the choice of op(i+1)i

(and so on λp) with i = 0, · · · , k but is independent of the choice of w+(i+1)i etc.

This is a consequence of the proof of Theorem 6.5. (We omit the detail of thispoint. See Remark 8.1.15 (3) [FOOO09].) Therefore the orientation in Theorem6.7 is independent of the choice of anchors.

Remark 6.8. In order to give an orientation of M(L; �p;B), we have to takethe moduli parameters of marked points and the action of the automorphism groupinto account. We also treat the intersection point pi(i−1) as if it is a chain of

codimension μ([pi(i−1), w+i(i−1)];λ(i−1)i) in a similar way to Chapter 8, Section 8.5

in [FOOO09].

33


7. Floer chain complex

In this subsection, we will describe construction of the boundary map. We alsomention some (minor) modification needed in its construction in the context withanchored Lagrangian submanifolds.

Let (Li, γi) i = 0, 1 be anchored Lagrangian submanifolds. We write E =((L0, γ0), (L1, γ1)). Let p, q ∈ L0∩L1 be admissible intersection points. We definedthe set π2(p, q) = π2((L0, L1), (p, q)) in Section 3. We also defined π2(�01; p) there.We now define:

Definition 7.1. CF ((L1, γ1), (L0, γ0)) is a free R module over the basis [p, w]where p ∈ L0 ∩ L1 is an admissible intersection points and [w] ∈ π2(�01; p).

Here R is a ground ring such as Q, Z, Z2, C or R. (The choice Z or Z2 requiressome additional conditions.)

Remark 7.2. We remark that the set of [p, w] where p is the admissible inter-section point is identified with the set of the critical point of the action functional Adefined on the Novikov covering space of Ω(L0, L1; �01). The group Π(L0, L1; �01)defined in Section 5 acts freely on it so that the quotient space is the set of admis-sible intersection points.

We next take a grading λi to (Li, γi) as in Subsection 5.2. It induces a gradingof [p, w] given by μ([p, w];λ01), which gives the graded structure on CF (L1, L0; �01)

CF (L1, L0; �01) =⊕k

CF k(L1, L0;λ01)

where CF k(L1, L0;λ01) = spanR{[p, w] | μ([p, w];λ01) = k}.For given B ∈ π2(p, q), we denote by Map(p, q;B) the set of such w’s in class

B.We summarize the extra structures added in the discussion of Floer homology

for the anchored Lagrangian submanifolds in the following

Situation 7.3. We assume that (L0, L1) is a relatively spin pair. We considera pair (L0, γ0), (L1, γ1) of anchored Lagrangian submanifolds and the base path�01 = γ0 ∗ γ1. We fix a grading λi of γi for i = 0, 1, which in turn induce a gradingof �01, λ01 = λ0 ∗λ1. We also fix an orientation op of Index ∂λp

for each p ∈ L0∩L1

as in Theorem 6.5.

We sometime do not explicitly write these extra data in our notations belowas long as there is no danger of confusion.

Let us consider Situation 7.3. Orientations of the Floer moduli spaceM(p, q;B)is induced by Theorem 6.5. Using virtual fundamental chain technique [FOn],Appendix A.1 [FOOO09] we can take a system of multisections and obtain asystem of rational numbers n(p, q;B) = #(M(p, q;B)) whenever the virtual di-mension of M(p, q;B) is zero. Finally we define the Floer ‘boundary’ map ∂ :CF (L1, L0; �01) → CF (L1, L0; �01) by the sum

(7.1) ∂([p, w]) =∑

q∈L0∩L1

∑B∈π2(p,q)

n(p, q;B)[q, w#B].

By Remark 7.2, CF (L1, L0; �01) carries a natural Λ(L0, L1; �01)-module structureand CF k(L1, L0;λ01) a Λ(0)(L0, L1; �01)-module structure where

Λ(0)(L0, L1; �01) ={∑

ag[g] ∈ Λ(L0, L1; �01)∣∣∣μ([g]) = 0

}.

34


We define

(7.2) C(L1, L0; �01) = CF (L1, L0; �01)⊗Λ(L0,L1;�01) Λnov

where we use the embedding (2.7).We write the Λnov module (7.2) also as

C((L1, γ1), (L0, γ0); Λnov).

Definition 7.4. We define the energy filtration FλCF ((L1, γ1), (L0, γ0)) ofthe Floer chain complex CF (L1, γ1), (L0, γ0)) (here λ ∈ R) such that [p, w] is inFλCF ((L1, γ1), (L0, γ0)) if and only if A([p, w]) ≥ λ.

This filtration also induces a filtration on (7.2).

Remark 7.5. We remark that this filtration depends (not only of the homotopyclass of) but also of γi itself.

It is easy to see the following from the definition of ∂ above:

Lemma 7.6.

∂(FλCF ((L1, γ1), (L0, γ0)) ⊆ FλCF ((L1, γ1), (L0, γ0))

).

8. Obstruction and A∞ structure

Let (L0, L1) be a relatively spin pair with L0 intersecting L1 transversely andfix a (M, st)-relatively spin structure for the pair (L0, L1).

According to the definition (7.1) of the map ∂, we have the formula for itsmatrix coefficients

(8.1) 〈∂∂[p, w], [r, w#B]〉 =∑

q∈L0∩L1

∑B=B1#B2∈π2(p,r)

n(p, q;B1)n(q, r;B2)Tω(B)

where B1 ∈ π2(p, q) and B2 ∈ π2(q, r).To prove, ∂∂ = 0, one needs to prove 〈∂∂[p, w], [r, w#B]〉 = 0 for all pairs

[p, w], [r, w#B]. On the other hand it follows from definition that each summand

n(p, q;B1)n(q, r;B2)Tω(B) = n(p, q;B1)T

ω(B1)n(q, r;B2)Tω(B2)

and the coefficient n(p, q;B1)n(q, r;B2) is nothing but the number of broken tra-jectories lying in M(p, q;B1)#M(q, r;B2). This number is nonzero in the generalsituation we work with.

To handle the problem of obstruction to ∂ ◦ ∂ = 0 and of bubbling-off discsin general, a structure of filtered A∞ algebra (C,m) with non-zero m0-term isassociated to each Lagrangian submanifold L [FOOO00, FOOO09].

8.1. A∞ algebra. In this subsection, we review the notion and constructionof filtered A∞ algebra associated to a Lagrangian submanifold. In order to makethe construction consistent to one in the last section, where Λ(L0, L1; �01) is usedfor the coefficient ring rather than the universal Novikov ring, we rewrite themusing smaller Novikov ring Λ(L) which we define below. Let L be a relatively spinLagrangian submanifold. We have a homomorphism

(E, μ) : H2(M,L;Z) → R× Z

35


where E(β) = β ∩ [ω] and μ is the Maslov index homomorphism. We put g ∼ g′ forg, g′ ∈ H2(M,L; : Z) if E(g) = E(g′) and μ(g) = μ(g′). We write Π(L) the quotientwith respect to this equivalence relation. It is a subgroup of R× Z. We define

Λ(L) ={∑

cg[g]∣∣∣ g ∈ Π(L), cg ∈ R,E(g) ≥ 0,

∀E0 #{g | cg �= 0, E(g) ≤ E0} < ∞}

There exists an embedding Λ(L) → Λ0,nov, defined by [g] �→ eμ(g)/2TE(g).

Let C be a graded R-module and CF = C⊗RΛ(L). Here and hereafter we usesymbol CF for the modules over Λ(L) or Λ(L0, L1) and C for the modules over theuniversal Novikov ring.

We denote by CF [1] its suspension defined by CF [1]k = CF k+1. We denoteby deg(x) = |x| the degree of x ∈ C before the shift and deg′(x) = |x|′ that afterthe degree shifting, i.e., |x|′ = |x| − 1. Define the bar complex B(CF [1]) by

Bk(CF [1]) = (CF [1])k⊗, B(CF [1]) =∞⊕k=0

Bk(CF [1]).

Here B0(CF [1]) = R by definition. The tensor product is taken over Λ(L). Weprovide the degree of elements of B(CF [1]) by the rule

(8.2) |x1 ⊗ · · · ⊗ xk|′ :=k∑

i=1

|xi|′ =k∑

i=1

|xi| − k

where | · |′ is the shifted degree. The ring B(CF [1]) has the structure of gradedcoalgebra.

Definition 8.1. The structure of strict filtered A∞ algebra over Λ(L) is asequence of Λ(L) module homomorphisms

mk : Bk(CF [1]) → CF [1], k = 1, 2, · · · ,of degree +1 such that the coderivation d =

∑∞k=1 mk satisfies dd = 0, which is

called the A∞-relation. Here we denote by mk : B(CF [1]) → B(CF [1]) the uniqueextension of mk as a coderivation on B(CF [1]). A filtered A∞ algebra is an A∞algebra with a filtration for which mk are continuous with respect to the inducenon-Archimedean topology.

In particular, we have m1m1 = 0 and so it defines a complex (CF,m1). Wedefine the m1-cohomology by

(8.3) H(CF,m1) = Kerm1/Imm1.

A filtered A∞ algebra is defined in the same way, except that it also includes

m0 : R → B(CF [1]).

The first two terms of the A∞ relation for a A∞ algebra are given as

m1(m0(1)) = 0(8.4)

m1m1(x) + (−1)|x|′m2(x,m0(1)) +m2(m0(1), x) = 0.(8.5)

In particular, for the case m0(1) is nonzero, m1 will not necessarily satisfy theboundary property, i.e., m1m1 �= 0 in general.

36


Remark 8.2. Here we use the Novikov ring Λ(L). In Section 1.2 [FOOO09],we defined a filtered A∞ algebra over the universal Novikov ring Λ0,nov. A filteredA∞ algebra over Λ(L) induces one over Λ0,nov in an obvious way. On the otherhand, an appropriate gap condition is needed for a filtered A∞ algebra over Λ0,nov

to induce one over Λ(L).

We now describe the A∞ operators mk in the context of A∞ algebra of La-grangian submanifolds. For a given compatible almost complex structure J , con-sider the moduli space of stable maps of genus zero

Mk+1(β;L) = {(w, (z0, z1, · · · , zk)) | ∂w = 0, zi ∈ ∂D2, [w] = β in π2(M,L)}/ ∼

where ∼ is the conformal reparameterization of the disc D2. We require thatz0, · · · , zk respects counter clockwise cyclic order of S1. (We wrote this modulispace Mmain

k+1 (β;L) in Section 2.1 [FOOO09]. The symbol ‘main’ indicates thecompatibility of z0, · · · , zk, with counter clockwise cyclic order. We omit this sym-bol in this paper since we always assume it.)

Mk+1(β;L) has a Kuranishi structure and its dimension is given by

(8.6) n+ μ(β)− 3 + (k + 1) = n+ μ(β) + k − 2.

Now let [P1, f1], · · · , [Pk, fk] ∈ C∗(L;Q) be k smooth singular simplices of L. (Herewe denote by C(L;Q) a suitably chosen countably generated cochain complex ofsmooth singular chains of L.) We put the cohomological grading degPi = n−dimPi

and consider the fiber product

ev0 : Mk+1(β;L)×(ev1,··· ,evk) (P1 × · · · × Pk) → L.

A simple calculation shows that the expected dimension of this chain is given by

n+ μ(β)− 2 +∑k

j=1(dimPj + 1− n) or equivalently we have the degree

deg[Mk+1(β;L)×(ev1,··· ,evk) (P1 × · · · × Pk), ev0

]=

n∑j=1

(degPj − 1) + 2− μ(β).

For each given β ∈ π2(M,L) and k = 0, · · · , we define m1,0(P ) = ±∂P and

(8.7)mk,β(P1, · · · , Pk) =

[Mk+1(β;L)×(ev1,··· ,evk) (P1 × · · · × Pk), ev0

]

∈ C(L;Q)

(More precisely we regard the right hand side of (8.7) as a smooth singular chain bytaking appropriate multi-valued perturbation (multisection) and choosing a simpli-cial decomposition of its zero set.)

We put

CF (L) = C(L;Q) ⊗Q Λ(L).

We define mk : BkCF (L)[1] → BkCF [1] by

mk =∑

β∈π2(M,L)

mk,β ⊗ [β].

Then it follows that the map mk : BkCF (L)[1] → CF (L)[1] is well-defined, hasdegree 1 and continuous with respect to non-Archimedean topology. We extendmk as a coderivation mk : BCF [1] → BCF [1] where BCF (L)[1] is the completionof the direct sum ⊕∞

k=0BkCF (L)[1] where BkCF (L)[1] itself is the completion of

37


CF (L)[1]⊗k. BCF (L)[1] has a natural filtration defined similarly as Definition 7.4.Finally we take the sum

d =∞∑k=0

mk : BCF (L)[1] → BCF (L)[1].

We then have the following coboundary property:

Theorem 8.3. Let L be an arbitrary compact relatively spin Lagrangian sub-

manifold of an arbitrary tame symplectic manifold (M,ω). The coderivation d is

a continuous map that satisfies the A∞ relation dd = 0, and so (CF (L),m) is afiltered A∞ algebra over Λ(L).

We putC(L; Λ0,nov) = CF (L) ⊗Λ(L) Λ0,nov

on which a filtered A∞ structure on C(L; Λ0,nov) (over the ring Λ0,nov) is induced.This is the filtered A∞ structure given in Theorem A [FOOO09]. The proof is thesame as that of Theorem A [FOOO09].

In the presence of m0, m1m1 = 0 no longer holds in general. This leads to con-sider deforming Floer’s original definition by a bounding cochain of the obstructioncycle arising from bubbling-off discs. One can always deform the given (filtered) A∞algebra (CF (L),m) by an element b ∈ CF (L)[1]0 by re-defining the A∞ operatorsas

mbk(x1, · · · , xk) = m(eb, x1, e

b, x2, eb, x3, · · · , xk, e

b)

and taking the sum db =∑∞

k=0 mbk. This defines a new filtered A∞ algebra in

general. Here we simplify notations by writing

eb = 1 + b+ b⊗ b+ · · ·+ b⊗ · · · ⊗ b+ · · · .Note that each summand in this infinite sum has degree 0 in CF (L)[1] and convergesin the non-Archimedean topology if b has positive valuation, i.e., v(b) > 0. (SeeSection 2 for the definition of v.)

Proposition 8.4. For the A∞ algebra (CF (L),mbk), m

b0 = 0 if and only if b

satisfies

(8.8)

∞∑k=0

mk(b, · · · , b) = 0.

This equation is a version of Maurer-Cartan equation for the filtered A∞ algebra.

Definition 8.5. Let (CF (L),m) be a filtered A∞ algebra in general andBCF (L)[1] be its bar complex. An element b ∈ CF (L)[1]0 = CF (L)1 is calleda bounding cochain if it satisfies the equation (8.8) and v(b) > 0. We denote by

M(L; Λ(L)) the set of bounding cochains.

In general a given A∞ algebra may or may not have a solution to (8.8). In ourcase we define:

Definition 8.6. A filtered A∞ algebra (CF (L),m) is called unobstructed overΛ(L) if the equation (8.8) has a solution b ∈ CF (L)[1]0 = CF (L)1 with v(b) > 0.

One can define the notion of homotopy equivalence between two boundingcochains and et al as described in Chapter 4 [FOOO09]. We denote byM(L; Λ(L))the set of equivalence classes of bounding cochains of L.

38


Remark 8.7. In Definition 8.5 above we consider bounding cochain contained

in CF (L) ⊂ C(L; Λ0) only. This is the reason why we write M(L; Λ(L)) in place

of M(L). (The latter is used in [FOOO09].)

8.2. A∞ bimodule. Suppose we are in Situation 7.3. Once the A∞ algebrais attached to each Lagrangian submanifold L, we then construct a structure offiltered A∞ bimodule on the module CF ((L1, γ1), (L0, γ0)), which was introducedin Section 7 as follows. This filtered A∞ bimodule structure is by definition is afamily of operators

nk1,k0: Bk1

(CF (L1)[1]) ⊗Λ(L1) CF ((L1, γ1), (L0, γ0)) ⊗Λ(L0) Bk0(CF (L′)[1])

→ CF ((L1, γ1), (L0, γ0))

for k0, k1 ≥ 0. Here the left hand side is defined as follows: It is easy to see thatthere are embeddings Λ(L0) → Λ(L0, L1; �01), Λ(L1) → Λ(L0, L1; �01). Therefore aΛ(L0, L1; �01) module CF ((L1, γ1), (L0, γ0)) can be regarded both as Λ(L0) moduleand Λ(L1) module. Hence we can take tensor product in the left hand side. (⊗Λ(Li)

is the completion of this algebraic tensor product.) The left hand side then becomesa Λ(L0, L1; �01) module, since the rings involved are all commutative.

We briefly describe the definition of nk1,k0. A typical element of the tensor

product

Bk1(CF (L1)[1])⊗Λ(L1) CF ((L1, γ1), (L0, γ0)) ⊗Λ(L0) Bk0

(CF (L0)[1])

has the form

P1,1 ⊗ · · · ,⊗P1,k1⊗ [p, w]⊗ P0,1 ⊗ · · · ⊗ P0,k0

with p ∈ L0 ∩ L1 being an admissible intersection point. Then the image nk0,k1

thereof is given by∑q,B

Tω(B)eμ(B)/2#(M(p, q;B;P1,1, · · · , P1,k1;P0,0, · · · , P0,k0

)) [q, B#w].

Here B denotes homotopy class of Floer trajectories connecting p and q, the sum-mation is taken over all [q, B] with

dimM(p, q;B;P1,1, · · · , P1,k1;P0,1, · · · , P0,k0

) = 0,

and # (M(p, q;B;P1,1, · · · , P1,k1;P0,1, · · · , P0,k0

)) is the ‘number’ of elements in the‘zero’ dimensional moduli space M(p, q;B;P1,1, · · · , P1,k1

;P0,1, · · · , P0,k0). Here

the moduli space M(p, q;B;P1,1, · · · , P1,k1;P0,1, · · · , P0,k0

) is the Floer modulispace M(p, q;B) cut-down by intersecting with the given chains P1,i ⊂ L1 andP0,j ⊂ L0. (See Section 3.7 [FOOO09].) An orientation on this moduli space isinduced by o[p,w], o[q,w′], which we obtained by Theorem 6.5.

Theorem 8.8. Let ((L0, γ0), (L1, γ1)) be a pair of anchored Lagrangian sub-manifolds. Then the family {nk1,k0

} defines a left (CF (L1),m) and right (CF (L0),m)filtered A∞-bimodule structure on CF ((L1, γ1), (L0, γ0)).

See Section 3.7 [FOOO09] for the definition of filtered A∞ bimodules. (In[FOOO09] the case of universal Novikov ring as a coefficient is considered. It iseasy to modify to our case of Λ(L0, L1) coefficient.) The proof of Theorem 8.8 isthe same as that of Theorem 3.7.21 [FOOO09].

39


In the case where both L0, L1 are unobstructed, we can carry out this defor-mation of n using bounding cochains b0 and b1 of CF (L0) and CF (L1) respec-tively, in a way similar to mb. Namely we define δb1,b0 : CF ((L1, γ1), (L0, γ0)) →CF ((L1, γ1), (L0, γ0)) by

δb1,b0(x) =∑k1,k0

nk1,k0(b⊗k1

1 ⊗ x⊗ b⊗k00 ) = n(eb1 , x, eb0).

We can generalize the story to the case where L0 has clean intersection with L1,especially to the case L0 = L1. In the case L0 = L1 we have nk1,k0

= mk0+k1+1. Soin this case, we have δb1,b0(x) = m(eb1 , x, eb0).

We define Floer cohomology of the pair (L0, γ0, λ0), (L1, γ1, λ1) by

HF ((L1, γ1, b1), (L0, γ0, b0)) = Ker δb1,b0/ Im δb1,b0 .

This is a module over Λ(L0, L1; �01).

Theorem 8.9. HF ((L1, γ1, b1), (L0, γ0, b0)) ⊗Λ(L0,L1) Λnov is invariant un-der the Hamiltonian isotopies of L0 and L1 and under the homotopy of boundingcochains b0, b1.

The proof is the same as the proof of Theorem 4.1.5 [FOOO09] and so omitted.

8.3. Products. Let L = (L0, L1, · · · , Lk) be a chain of compact Lagrangiansubmanifolds in (M,ω) that intersect pairwise transversely without triple intersec-tions.

Let �z = (z0k, zk(k−1), · · · , z10) be a set of distinct points on ∂D2 = {z ∈ C ||z| = 1}. We assume that they respect the counter-clockwise cyclic order of ∂D2.The group PSL(2;R) ∼= Aut(D2) acts on the set in an obvious way. We denote by

Mmain,◦k+1 be the set of PSL(2;R)-orbits of (D2, �z).In this subsection, we consider only the case k ≥ 2 since the case k = 1 is

already discussed in the last subsection. In this case there is no automorphism onthe domain (D2, �z), i.e., PSL(2;R) acts freely on the set of such (D2, �z)’s.

Let pj(j−1) ∈ Lj ∩ Lj−1 (j = 0, · · · k), be a set of intersection points.

We consider the pair (w; �z) where w : D2 → M is a pseudo-holomorphic mapthat satisfies the boundary condition

w(zj(j−1)z(j+1)j) ⊂ Lj ,(8.9a)

w(z(j+1)j) = p(j+1)j ∈ Lj ∩ Lj+1.(8.9b)

We denote by M◦(L, �p) the set of such ((D2, �z), w).We identify two elements ((D2, �z), w), ((D2, �z′), w′) if there exists ψ ∈ PSL(2;R)

such that w ◦ ψ = w′ and ψ(z′j(j−1)) = zj(j−1). Let M◦(L, �p) be the set of equiva-

lence classes. We compactify it by including the configurations with disc or spherebubbles attached, and denote it by M(L, �p). Its element is denoted by ((Σ, �z), w)where Σ is a genus zero bordered Riemann surface with one boundary components,�z are boundary marked points, and w : (Σ, ∂Σ) → (M,L) is a bordered stable map.

We can decompose M(L, �p) according to the homotopy class B ∈ π2(L, �p) ofcontinuous maps satisfying (8.9a), (8.9b) into the union

M(L, �p) =⋃

B∈π2(L;p)

M(L, �p;B).

40


In the case we fix an anchor γi to each of Li and put E = ((L0, γ0), · · · , (Lk, γk)),we consider only admissible classes B and put

M(E , �p) =⋃

B∈πad2 (E;p)

M(E , �p;B).

Theorem 8.10. Let L = (L0, · · · , Lk) be a chain of Lagrangian submanifoldsand B ∈ π2(L; �p). Then M(L, �p;B) has an oriented Kuranishi structure (withboundary and corners). Its (virtual) dimension satisfies

(8.10) dimM(L, �p;B) = μ(L, �p;B) + n+ k − 2,

where μ(L, �p;B) is the polygonal Maslov index of B.

Proof. We consider the operator ∂w;L in (6.11). It is easy to see that

(8.11) Index ∂w;L + k − 2 = dimM(L, �p;B).

In fact k − 2 in the left hand side is the dimension of Mmain,◦k+1 /PSL(2;R).

We next consider the Fredholm operator (6.13). By (8.11), Lemma 5.12 andindex sum formula, we have

(8.12) Index ∂w;L − μ(L, �p;B) = Index of (6.13).

We remark that the operator (6.13) is a Cauchy-Riemann operator of the trivial Cn

bundle on D2 with boundary condition determined by a certain loop in Lag(Cn, ω).By construction it is easy to see that this loop is homotopic to a constant loop.Therefore, the index of (6.13) is n. Theorem 8.10 follows. �

We next take graded anchors (γi, λi) to each Li and fix the data as in Situation7.3. We assume that B is admissible and write B = [w−

01]#[w−12]# · · ·#[w−

k0] as

in Definition 3.2. We put w+(i+1)i(s, t) = w−

i(i+1)(1 − s, t) as in (5.9). We also put

w+k0(s, t) = w+

0k(s, 1− t). ([w+k0] ∈ π1(�k0; pk0).) We also put λk0(t) = λ0k(1− t).

Lemma 8.11. If dimM(L, �p;B) = 0, we have

(8.13) (μ([pk0, w+k0];λ0k)− 1) = 1 +

k∑i=1

(μ([pi(i−1), w+i(i−1)];λ(i−1)i)− 1).

Proof. Lemma 5.12 and Theorem 8.10 implies

k∑i=0

μ([p(i+1)i, w+(i+1)i];λi(i+1)) = n+ k − 2

in the case dimM(L, �p;B) = 0. By Lemma 5.8 we have

μ([p0k, w+0k];λk0) = −μ([pk0, w

+k0];λ0k) + n.

Substituting this into the above identity and rearranging the identity, we obtainthe lemma. �

Using the case dimM(L, �p;B) = 0, we define the k-linear operator

mk : CF ((Lk, γk), (Lk−1, γk−1))⊗. . .⊗CF ((L1, γ1), (L0, γ0)) → CF ((Lk, γk), (L0, γ0))

as follows:

(8.14)mk([pk(k−1), w

+k(k−1)],[p(k−1)(k−2), w

+(k−1)(k−2)], · · · , [p10, w

+10]))

=∑

#(Mk+1(L; �p;B)) [pk0, w+k0]).

41


Here the sum is over the basis [pk0, w+k0] of CF ((Lk, γk), (L0, γ0)), where �p =

(p0k, pk(k−1), · · · , p10), B is as in Definition 3.2, and w+(i+1)i(s, t) = w−

i(i+1)(1− s, t).

The formula (8.13) implies that mk above has degree one.In general the operator mk above does not satisfy the A∞ relation by the

same reason as that of the case of boundary operators (see Section 7). We needto use bounding cochains bi of Li to deform mk in the same way as the case ofA∞-bimodules (Subsection 8.2), whose explanation is now in order.

Let m0, · · · ,mk ∈ Z≥0 and Mm0,··· ,mk(L, �p;B) be the moduli space obtained

from the set of ((D2, �z), (�z(0), · · · , �z(k)), w)) by taking the quotient by PSL(2,R)-action and then by taking the stable map compactification as before. Here z(i) =

(z(i)1 , · · · , z(i)ki

) and z(i)j ∈ z(i+1)izi(i−1) such that z(i+1)i, z

(i)1 , · · · , z(i)ki

, zi(i−1) re-spects the counter clockwise cyclic ordering.

((D2, �z), (�z(0), · · · , �z(k)), w)) �→ (w(z(0)1 ), · · · , w(z(k)mk

))

induces an evaluation map:

ev = (ev(0), · · · , ev(k)) : Mm0,··· ,mk(L, �p;B) →

k∏i=0

Lmii .

Let P(i)j be smooth singular chains of Li and put

�P (i) = (P(i)1 , · · · , P (i)

mi),

��P = (�P (0), · · · , �P (k))

We then take the fiber product to obtain:

Mm0,··· ,mk(L, �p;

��P ;B) = Mm0,··· ,mk(L, �p;B)×ev

��P.

We use this to define

mk;m0,··· ,mk: Bmk

(CF (Lk))⊗ CF ((Lk, γk), (Lk−1, γk−1))⊗ · · ·⊗ CF ((L1, γ1), (L0, γ0))⊗Bm0

(CF (L0)) → CF ((Lk, γk), (L0, γ0))

by

mk;m0,··· ,mk(�P (k), [pk(k−1), w

+k(k−1)], · · · , [p10, w

+10],

�P (0))

=∑

#(Mk+1(L; �p;��P ;B)) [pk0, wk0].

Finally for each given bi ∈ CF (Li)[1]0 (bi ≡ 0 mod Λ+), �b = (b0, · · · , bk), and

xi ∈ CF ((Li, γi), (Li−1, γi−1)), we put

(8.15) mbk(xk, · · · , x1) =

∑m0,··· ,mk

mk;m0,··· ,mk(bmk

k , xk, bmk−1

k−1 , · · · , x1, bm00 ).

Theorem 8.12. If bi satisfies the Maurer-Cartan equation (8.8) then mbk in

(8.15) satisfies the A∞ relation

(8.16)∑

k1,k2,i

(−1)∗mbk1(xk, · · · ,m

bk2(xk−i−1, · · · , xk−i−k2

), · · · , x1) = 0

where we take sum over k1+k2 = k+1, i = −1, · · · , k−k2. (We write mk in place

of mbk in (8.16).) The sign ∗ is ∗ = i+ deg xk + · · ·+ deg xk−i.

The non-anchored version is proved in Theorem 4.17 [Fu2]. In order to trans-late it to the anchored version we only need to show the following.

42


Lemma 8.13. Let E = ((L0, γ0), · · · , (Li, γi), · · · , (Lj , γj), · · · , (Lk, γk)), �p =(pk(k−1), · · · , pj(j−1), · · · , pi(i−1), · · · p10) and B ∈ πad

2 (E ; �p) be admissible. Sup-pose that the sequence ui ∈ M(L, �p;B) converges to an element in the productM(L′, �p ′;B1)×M(L′′, �p ′′;B2) where

L′ = (L0, · · · , Li, Lj , · · · , Lk), L′′ = (Li, Li+1, · · · , Lj)

and

�p ′ = (p0k, pk(k−1), · · · , p(j+1)j , pji, pi(i−1), · · · , p10), �p ′′ = (pij , pj(j−1) · · · , p(i+1)i)

for some pij = pji ∈ Li ∩ Lj.Then B1, B2 are E ′, E ′′ admissible, respectively.

Proof. For simplicity of notations, we only consider the case k = 3, i = 1,j = 3. Let ua ∈ M(L, �p;B) which converges to u∞ = (u∞,1, u∞,2) where

u∞,1 ∈ M((L0, L1, L3), (p03, p31, p10);B1),

u∞,2 ∈ M((L1, L2, L3), (p13, p32, p21);B2),

and p13 = p31. By definition of E-admissibility of B, there exist homotopy classes[w−

i(i+1)] ∈ π2(Li(i+1); �i(i+1)) for 0 ≤ i ≤ 3 such that B = [w−01#w−

12#w−23#w−

30].

To prove the required admissibility of B1, B2, we need to prove the existence ofhomotopy classes [w−

13] ∈ π2(�13; p13) and [w−31] ∈ π2(�31; p31) such that

[w−01#w−

13#w−30] = B1, [w

−12#w−

23#w−31] = B2

and [w−13#w−

31] = [p13, p13] where p13 is the constant map to p13. In fact, we willselect both homotopy classes to be that of p13 = p31.

Exploiting E-admissibility of B, we can take the sequences of points xa ∈ua(D

2) ⊂ M , of paths γa : [0, 1] → M and γa,i : [0, 1] → ua(D2) such that

γa(0) = y, γa(1) = xa,

γa,i(0) = xa, γa,i(1) ∈ Li

and that γa ∗ γa,i is homotopic to the given anchor γi.Deforming these choices further, we may use the convergence hypothesis to

achieve the following additional properties of xa, γa and γa,i’s:

(1) lima→∞ xa = p13,(2) lima→∞ γa,i(t) ≡ p13 for i = 1, 3,(3) γa,i converges to a path γ∞,i as a → ∞,(4) and γa converges to a path γ∞ as a → ∞.

From this and by construction of γi, γa,i, it is easy to see that B1 and B2 are((L0, γ∞,0), (L1, γ∞), (L3, γ∞)) admissible and ((L1, γ∞), (L2, γ2,∞), (L3, γ∞)) ad-missible respectively. In fact, since γ∞ is homotopic to γi for i = 1, 3, γ∞,j ishomotopic to γj for j = 0, 2, we can express

B1 = [w−01#p13#w−

30], B2 = [w−12#w−

23#p31].

This finishes the proof. �

We summarize the above discussion as follows:

Theorem 8.14. We can associate an filtered A∞ category to a symplectic man-ifold (M,ω) such that:

43


(1) Its object is ((L, γ, λ), b, sp) where (L, γ, λ) is a graded anchored Lagrangiansubmanifold, [b] ∈ M(CF (L)) is a bounding cochain and sp is a spinstructure of L.

(2) The set of morphisms is CF ((L1, γ1), (L0, γ0)).

(3) mbk are the operations defined in (8.15).

Remark 8.15. In Situation 7.3, beside the choices spelled out in ((L, γ, λ), b, sp),the choice of orientations op of Index ∂λp

is included. This choice in fact does notaffect the module structure CF ((L1, γ1), (L0, γ0)) up to isomorphism: if we take analternative choice o′p at p, then all the signs appearing in the operations mk thatinvolves [p, w] for some w will be reversed. Therefore [p, w] �→ −[p, w] gives therequired isomorphism.

Remark 8.16. In [Fu2], the filtered A∞ category is defined over Λ0,nov. Thesituation of Theorem 8.14 is slightly different in that Λ(L0, L1; �01) or Λ(L) areused as the coefficient rings and hence the coefficient rings vary depending on theobjects involved. It is easy to see that the notion of filtered A∞ category can begeneralized to this context.

We can also change the coordinate ring to Λnov by using the map [p, w] �→T

∫w∗ωeμ(w)/2〈p〉 (Subsection 5.1.3 [FOOO09]). The resulting filtered A∞ cate-

gory is still different from the non-anchored version in the case M is not simplyconnected.

Remark 8.17. In Theorem 8.14, we assume that our Lagrangian submanifoldL is spin. We can slightly modify the construction to accommodate the relativelyspin case as follows: We will construct the filtered A∞ category of ((M,ω), st)for each choice of st ∈ H2(M ;Z2). Its objects consist of ((L, γ, λ), b, sp) whereL satisfies w2(L) = i∗(st) and (L, γ, λ), b are as before. Finally sp is the stableconjugacy class of relative spin structure of L. (See Definition 8.1.5 [FOOO09] forits definition.) In this way we obtain a filtered A∞ category. The same remarkapplies to the non-anchored case.

The operations mk are compatible with the filtration. Namely we have

Proposition 8.18. If xi ∈ FλiCF ((Li, γi), (Li−1, γi−1)), then

mbk(xk, · · · , x1) ∈ FλCF ((Lk, γk), (L0, γ0))

where λ =∑k

i=1 λi.

Studying the behavior of filtration under the A∞ operations, one can define(higher-order) spectral invariants of Lagrangian Floer theory in a way similar tothe one carried out in [Oh2]. Then Proposition 8.18 implies a similar estimate asTheorem I(4) [Oh2]. This is a subject of future study.

9. Comparison between anchored and non-anchored versions

The anchored Lagrangian Floer theory presented in this paper is somewhatdifferent from the one developed in [FOOO00, FOOO09] (for one and two La-grangian submanifolds) [Fu2] (for 3 or more Lagrangian submanifolds) in severalpoints. In this section we examine their relationship and make some comments onsome aspects of their applications.

We first point out the following obvious fact:

44


Proposition 9.1. If M is simply connected the anchored version of Floer ho-mology is isomorphic to non-anchored version together with all of its multiplicativestructures.

We also remark that the way how we treat the orientation for the anchoredversion in Section 6 is actually the same as the one used for the non-anchoredversion in [FOOO09] and [Fu2].

9.1. Examples. We start with simple examples that illustrate some differencebetween the two.

We consider the symplectic manifold (T 2, dx∧dy), where T 2 = R2/Z2 and (x, y)is the standard coordinate of R2. Let L0 = {[x, 0] | x ∈ R}, L1 = {[x, 3x] | x ∈ R}.

L0 ∩ L1 = {[0, 0], [1/3, 0], [2/3, 0]}.

Let [1/2, 0] be the base point and take anchors

γ0(t) = [(1− t)/2, 0], γ01(t) = [(1− t)/2, 0],

of L0 and L1 respectively. It is easy to see that [0, 0] is ((L0, γ0), (L1, γ01)) ad-

missible. It is also easy to see by drawing pictures that [1/3, 0], [2/3, 0] are not((L0, γ0), (L1, γ

01)) admissible. The set of homotopy classes of anchors of L1 is

identified with Z and [k/3, 0] is ((L0, γ0), (L1, γ�1)) admissible if k ≡ � mod 3.

Here anchor γi1 is a concatenation of γ0

1 and t �→ [it/3, 0]. We also remark thatΠ(L0),Π(L1),Π(L0, L1) are trivial. Therefore Λ(L0) = Λ(L1) = Λ(L0, L1) = Q.Thus we have

HF ((L1, γi1), (L0, γ0)) ∼= Q

for any choice of anchor γi1.

Remark 9.2. In this subsection, we always take 0 as the bonding cochain band we omit it from the notation of Floer cohomology.

On the other hand we have

HF (L1, L0; Λnov) ∼= Λ⊕3nov.

It is easy to see that π0(Ω(L0, L1)) consists of 3 elements, which we denote �i01(i = 0, 1, 2). Moreover [γ0 ∗ γj

1] = �i01 with i ≡ j mod 3.Hence we have the decomposition

HF (L1, L0; Λnov) ∼=2⊕

i=0

HF (L1, L0; �i01; Λnov).

This is the decomposition given in Remark 3.7.46 [FOOO09]. (We note that wehave the isomorphism

HF (L1, L0; �i01; Λnov) ∼= HF ((L1, γ

i1), (L0, γ0))⊗ Λnov).

We next consider the same T 2 and

L0 = {[0, y] | y ∈ R}, L1 = {[x, 0] | x ∈ R}.

Then L0 ∩ L1 consists of one point [0, 0]. Therefore

HF ((L1, γ1), (L0, γ0)) ∼= Q

45


for any anchor γ0 and γ1. In fact Ω(L0, L1) is connected in this case. We nextconsider the third Lagrangian submanifold L2 = {[x,−x] | x ∈ R}. It is also easyto see that

HF (L2, Li; Λnov) ∼= HF ((L2, γ2), (Li, γi))⊗ Λnov∼= Λnov,

for i = 0, 1 and any anchor γj of Lj .We take [0, 0] as base point and take anchors

γk0 (t) = [kt, 0], γk

1 (t) = [0, kt], γk2 (t) = [kt/2, kt/2].

of Li for i = 0, 1, 2. For each k, � ∈ Z and i, j (i, j ∈ {0, 1, 2}, i �= j) we haveHF ((Li, γ

ki ), (Lj , γ

�j))

∼= Q. Let xk�ij = [[0, 0], wij;k�] be its canonical generator.

Here [wij;k�] represents the unique element of π2(γki ∗ γ�

j , [0, 0]).Let xij = 〈[0, 0]〉 be also the canonical generator of the (non-anchored) Floer

homology HF (Li, Lj). (We refer readers to Section 10.2 of present paper andSubsection 5.1.3 [FOOO09] for the definition of 〈[0, 0]〉.)

Now the product m2 is described as follows:

Proposition 9.3. In the case of non-anchored version we have

(9.1) m2(x21, x10) =

(∑k∈Z

T k2/2

)x20.

In the anchored version we have

(9.2) m2(xm�21 , x

�k10) = xmk

20 .

Proof. We first remark that π2((L0, L1, L2), (p02, p21, p10)) ∼= Z. Moreovereach of the homotopy class is realized by holomorphic disc uniquely. This implies(9.1).

To prove (9.2). it suffices to see that for each γk0 , γ

�1, γ

m2 the set

(9.3) πadm2 ((L0, γ

k0 ), (L1, γ

�2), (L2, γ

m2 )), (p02, p21, p10))

of admissible class consists of one element. We will prove it below.Let B be an element of (9.3). We write B = [w−

01]#[w−12]#[w−

20] as in Definition3.2. Let R2 → T 2 be the universal covering. We lift anchors γk

0 , γ�1, γ

m2 to γk

0 , γ�1,

γm2 such that γk

0 (0) = γ�1(0) = γm

2 (0) = 0.We then lift w01 such that (a part of) its boundary is γk

0 and γ�1. We lift w12

and w20 in a similar way. We thus obtain a lift w of w. It is easy to see that the

boundary of w(D2) is contained in Lk0 ∪ L�

1 ∪ Lm2 where

Lk0 = {(k, y) | y ∈ R}, L�

1 = {(x, �) | x ∈ R}, Lm2 = {(x,m− x) | x ∈ R}.

Thus the admissible homotopy class of B is unique. �

Remark 9.4. We remark that (9.1) is the formula appearing in Kontsevich[Ko] where the homological mirror symmetry proposal first appeared. So it seemsthat the anchored version is not suitable for the application to mirror symmetry,when M is not simply connected. On the other hand, the anchored version is moreclosely related to the variational theoretical origin of Floer homology and so seemsmore suitable to study spectral invariant for example.

The above proof also implies the following:

46


Lemma 9.5. If B is (L0, γk0 ), (L1, γ

�2), (L2, γ

m2 )) admissible then

(9.4) B ∩ ω =(m− k − �)2

2.

Remark 9.6. In the case of Lemma 9.3 we obtain the non-anchored version bysumming up anchored versions appropriately. In general the non-anchored version isan appropriate sum of anchored versions. However the way of summing up anchoredversions to obtain the non-anchored one does not look so simple to describe.

9.2. Relationship with the grading of Lagrangian submanifolds. In[Fu2] the first named author followed the method of Seidel [Se1] (and Kontsevich)to define a grading of Floer cohomology. In this section we discuss its relation tothe formulation of Section 5.

We first briefly recall the notion of gradings in the sense of [Se1]. Consider thetangent space TpM and let Lag+(TpM) be the set of oriented Lagrangian subspaces.The union Lag+(M) := ∪p∈MLag+(TpM) forms a fiber bundle over M . If L is anoriented Lagrangian submanifold, the Gauss map p �→ TpL provides a canonicalsection of the restriction Lag+(M)|L → L. We denote the canonical section by sL.

We first consider the case (M,ω) with c1(M) = 0.The fundamental group of Lag+(TpM) is Z. The condition c1(M) = 0 is

equivalent to the condition that there exists a (global) Z fold covering Lag(M) ofLag+(M), which restricts to the universal covering on each fiber Lag+(TpM). (SeeLemma 2.6 [Fu2].)

The section sL lifts to a section s of Lag(M)|L if and only if the Maslov classμL ∈ H1(L;Z) of L is zero. (Recall if c1(M) = 0, then the Maslov class μL iswell-defined.) For each Lagrangian submanifold L with μL = 0 a lift s of sL is saidto be a grading of L. The pair (L, s) of Lagrangian submanifold L and grading s iscalled a graded Lagrangian submanifold.

Let (Li, si) be a graded Lagrangian submanifold. Then for p ∈ L0 ∩ L1 we

consider any path λ from s0(p) to s1(p) in Lag(TpM) and denote its projection toLag(TpM) by λ. Then we compute the intersection number of λ with the Maslovcycle Lag1(TpM ;TpL0) (relative to TpL0) to define a degree deg p ∈ Z for each

p ∈ L0 ∩ L1. This definition is independent of the choice of λ with λ(0) = s0(p),

λ(1) = s1(p). (See [Se1], [Fu2] for the details.)Now we explain how the grading λ of (L, γ) and the grading s of L are related

to each other. For this purpose, we fix, once and for all, an element Vy of Lag(TyM)which projects to Vy ∈ Lag(TyM) at the base point y in Definition 5.4.

First, we go from s to λ. We consider any anchored Lagrangian submanifold

(L, γ) with μL = 0. Let s be a grading of L. We take a section of λi of the pull-back

γ∗i (Lag(TM)) → [0, 1] such that

λi(0) = Vy, λi(1) = si(λi(1)).

Such path is unique up to homotopy because [0, 1] is contractible and so γ∗i Lag(TM)

is simply connected. We push it out and obtain a section λ in γ∗Lag(TM). In thisway, a graded Lagrangian submanifold (L, s) canonically determines a grading λof an anchored Lagrangian submanifold (L, γ). Namely (γ, λ) becomes a gradedanchor of L in the sense of Definition 5.4.

We remark that the path λ01 induced by these graded anchors lifts to λ01

joining s0(�01(0)) to s1(�01(1)).

47


We then define μ([p, w]) using this path λ01 as in Section 5.

Lemma 9.7. μ([p, w]) is independent of w. Moreover we have

(9.5) μ([p, w]) = deg(p).

Proof. Independence of the degree of w is a consequence of our assumptionthat Maslov index of L0, L1 are zero. Then the equality (9.5) follows easily bycomparing the definitions. We omit the detail. �

Thus the degree of [Fu2] and of this paper coincides under the assumption thatis Maslov index is 0.

Now we go from λ to s. For any given grading λ of (L, γ), we lift λ to a section

of γ∗Lag(TM) so that λ(0) = Vy. Then λ(1) is a lifting of λ(1) in Lag(Tγ(1)M .

Since the lifting of λ of λ is homotopically unique, λ(1) depends only on (L, γ) and

the fixed Vy. Therefore if μL = 0, then this determines a unique grading s of L

with s(γ(1)) = λ(1).The above discussion can be generalized to the case of Z2N -grading where the

Maslov index is divisible by 2N for some positive integer N rather than being zero.We leave this discussion to the readers.

10. Reduction of the coefficient ring and Galois symmetry

In this section, we study a reduction of the coefficient ring Λnov to the subringΛratnov or to the ring Q[[T 1/N ]][T−1].

Definition 10.1. We put

Λratnov =

{ ∞∑i=1

Tλieμi/2ai ∈ Λnov

∣∣∣λi ∈ Q

}.

We also define Λrat0,nov in a similar way.

This problem was studied by the first named author in [Fu3] in relation to theGalois symmetry of Floer cohomology over rational symplectic manifolds. Theorem2.4 in [Fu3] is Theorem 1.2 of the present paper. Its proof was given in [Fu3] asfar as mk (k = 0, 1) concerns. The case for k ≥ 2 was ‘left to the reader’ in [Fu3].In this section we give the detail of the discussion for the case k ≥ 2.

10.1. Rational versus BS-rational Lagrangian submanifolds. In thissubsection, we first clarify somewhat confusing usages of the terminology ‘rational’Lagrangian submanifolds in the literature (e.g. in [Oh1], [Fu3] etc.).

First we assume that there exists an integer mamb with mambω ∈ H2(M ;Z),i.e., (M,mambω) is integral or pre-quantizable. Then we choose a complex linebundle P with a unitary connection ∇ such that its curvature F∇ satisfies

(10.1) F∇ = 2π√−1mambω.

The pair (P,∇) is called a pre-quantum bundle of (M,mambω). We note that theconnection is flat on any Lagrangian submanifold by (10.1).

Definition 10.2. We say that a Lagrangian submanifold L is Bohr-Sommerfeldm-rational or simply BS m-rational if the image of the holonomy group (P|L,∇|L)is contained in {exp(2π

√−1kmamb/m) | k ∈ Z}. We say L is Bohr-Sommerfeld

48


rational or simply BS-rational if it is BS-rational for some m. We denote thesmallest such integer by mL.

When (P,∇) is trivial on L and mamb = 1, we call L a Bohr-Sommerfeld orbit.

By definition, it is easy to see mamb|mL for any BS-rational Lagrangian sub-manifold L.

Remark 10.3. In [Oh1] and [Fu3], the corresponding notion is called cyclicand just rational respectively. In this paper, we adopt the name Bohr-Sommerfeldrational which properly reflects the kind of rationality of holonomy group of thequantum line bundle (P,∇) relative to the Lagrangian submanifold.

A Lagrangian submanifold is often called (spherically) rational in literaturewhen {ω(π2(M,L))} ⊂ R is discrete. This is related to but not exactly the sameas the BS-rationality in the above definition.

Definition 10.4. Let (M,ω) be rational. We say that a Lagrangian subman-ifold is rational if Γω(L) := {ω(α) | α ∈ π2(M,L)} ⊂ R is discrete.

The following lemma shows the relationship between the BS rationality andthe rationality.

Lemma 10.5. If L is BS m-rational, then L is rational. Moreover Γω(L) ⊆{exp(2π

√−1kmamb/m) | k ∈ Z}.

The converse does not hold in general when L is not simply connected. Infact for the case M = (T 2, dx ∧ dy) and Lt = {[t, y] | y ∈ R}, (Here we regardT 2 = R

2/Z2.) every Lt is rational but only countably many of Lt’s are BS-rational.(The question on which Lt becomes BS-rational depends on the choice of pre-quantum bundle (P,∇). It is easy to see that we may choose the pre-quantumbundle so that Lt is BS-rational if and only if t ∈ Q.)

Using Lemma 10.5, it is easy to show that the coefficient ring of the filteredA∞ algebra C(L; Λ0,nov) can be reduced to the ring Q[[T 1/N ]][e, e−1] ⊂ Λrat

0,nov. Inparticular, if we define

(10.2) C(L; Λrat0,nov) = C(L;Q) ⊗Q Λrat

0,nov ⊂ C(L; Λ0,nov)

the operations mk induce a filtered A∞ structure on C(L; Λrat0,nov).

10.2. Reduction of the coefficient ring: non-anchored version. In thissubsection, we explain the way to reduce the coefficient ring of the filtered A∞category associated to a symplectic manifold to the ring Λrat

0,nov.

Let (M,ω) be a symplectic manifold with mambω ∈ H2(M ;Z). We fix a pre-quantum bundle (P,∇) of (M,mambω).

We fix any integer N ∈ Z+ and consider the set of BS N -rational Lagrangiansubmanifolds .

Definition 10.6. The N -rationalization of L (in (M,ω), (P,∇)) is a globalsection SL of P⊗N/mamb such that

‖SL‖ ≡ 1, ∇⊗N/mambSL = 0.

The following lemma is easy to show.

Lemma 10.7. L is BS N-rational if and only if it has an N-rationalization.

49


Let Li, i = 0, 1 be a pair of N BS-rational Lagrangian submanifolds. Let SLi

be N -rationalizations Li. For each p ∈ L0 ∩ L1, we define c(p) to be the smallestnonnegative real number such that

(10.3) exp(2πNc(p)√−1/mamb)SL0

(p) = SL1(p).

Proposition 10.8. Let length(L) = k + 1 ≥ 2. Define

(10.4) E′(B) :=

∫B

ω −k∑

i=0

c(p(i+1)i)

for B ∈ π2(L; �p). Then E′ has values in Z[1/N ] and satisfies the gluing ruleE′(B) = E′(B1) + E′(B2) whenever B = B1#B2 in the sense of Lemma 8.13.

Proof. Let w ∈ C∞(D2,L; �p) be a map such that [w] = B. We consider the

pull back bundle w∗P⊗N/mamb . Let γ : [0, 1] → ∂D2 be the map t �→ e2π√−1t.

Using SLiwe can construct a section s on γ∗w∗P⊗N/mamb such that

∇s = 0, s(1) = exp(2πNc(p)√−1/mamb)s(0).

Using the fact that the curvature of P⊗N/mamb is 2Nπω, we conclude E′(B) ∈Z[1/N ]. The gluing rule for E′ is obvious from its definition (10.10). �

We put

[[p]]�01 = T−∫w∗ω+c(p)[p, w]

∈ CF (L1, L0; �01)⊗Λ(L0,L1;�01) Λnov ⊂ C(L1, L0; Λnov).

Lemma 10.9. If we change the choice of the base point �01 then there existsk ∈ Z such that [[p]]�01 = ek[[p]]�′01 where e is the formal parameter encoding thedegree.

The proof of the lemma is easy and left to the reader.This lemma together with the discussion on the degree in the last section shows

that the role of the choice of the base point �01 is to fix a connected component ofΩ(L0, L1) and does not play an essential role in Lagrangian Floer theory.

Definition 10.10. We consider the Λratnov-submodule of C(L1, L0; Λnov) gener-

ated by [[p]] (p ∈ L0 ∩L1) and denote it by C(L1, L0; Λratnov). We define the module

C(L1, L0, ;Q[[T 1/N ]][T−1][e, e−1]) in the same way.

Remark 10.11. (1) We note that the Λratnov-sub-module C(L1, L0; Λ

ratnov) of

C(L1, L0; Λnov) depends on the choice of the rationalizations SLi. We omit them

from notation however.(2) In Subsection 5.1.3 [FOOO09] we put

〈p〉 = T−∫w∗ω[p, w].

Then C(L1, L0; Λ0,nov) is a free Λ0,nov module over the basis {〈p〉 | p ∈ L1 ∩ L0}.The difference of 〈p〉 from the present basis [[p]] is T−c(p). Namely

(10.5) 〈p〉 = T−c(p)[[p]]

〈p〉 coincides with the identity in Hom(Lp,Lp) which appeared in (2.30) [Fu2]and is used there to construct a filtered A∞ category.

50


We now consider the operator

mk;m0,··· ,mk: Bmk

(C(Lk; Λ0,nov))⊗ C(Lk, Lk−1; Λnov)⊗ · · ·⊗ C(L1, L0; Λnov)⊗Bm0

(C(L0; Λ0,nov)) → C(Lk, L0; Λnov)

Proposition 10.12. The image of

Bmk(C(Lk; Λ

rat0,nov))⊗ C(Lk, Lk−1; Λ

ratnov)⊗ · · ·

⊗ C(L1, L0; Λrat0,nov)⊗Bm0

(C(L0; Λrat0,nov))

by mk;m0,··· ,mkis in C(Lk, L0; Λ

ratnov).

The same conclusion holds for Q[[T 1/N ]][T−1][e, e−1].

Proof. Let B ∈ π2(L, �p) and M(L, �p;B) be as in Section 7.For simplicity, we will prove the proposition for the case m0 = · · · = mk = 0.

Let 〈pij〉 = T−∫w∗

ijω[pij , wij ]. By (8.14), we have

(10.6) 〈mk(〈pk(k−1)〉, · · · , 〈p10〉), 〈pk0〉〉 =∑

B∈π2(L,p)

TB∩ωeμ(B)/2#M(L, �p;B).

Here the left hand side denotes the 〈pk0〉-coefficient of mk(〈pk(k−1)〉, · · · , 〈p10〉).Therefore by (10.5) we have the matrix coefficients

(10.7)

〈mk([[pk(k−1)]], · · · , [[p10]]), [[pk0]]〉

=∑

B∈π2(L,p)

TB∩ω−∑k

i=0 c(p(i+1)i)eμ(B)/2#M(L, �p;B)

for k ≥ 1. Since B ∩ ω −∑k

i=0 c(p(i+1)i) = E′(B) is rational by Proposition 10.8,

the right hand side of (10.7) lies in C(Lk, L0; Λratnov) as required. �

Now we are ready to wrap up the proof of Theorem 1.2. By the assumptionc1(M) = 0 and vanishing of Maslov indices of Lagrangian submanifolds, all La-grangian submanifolds in the discussion below carry a grading s. We just denote sfor s below to simplify the notation.

For each given N , with mamb|N , we construct a filtered A∞ category overQ[[T 1/N ]][T−1]. Its object is (L, sp, b, s, SL) where L is a BS N -rational Lagrangiansubmanifold sp its spin structure, s a grading, b is a bounding cochain, and SL isN -rationalization. We assume that b ∈ C1(L;Q[[T 1/N ]]).

For two such objects we obtain a Q[[T 1/N ]][T−1] module

C(L1, L0;Q[[T 1/N ]][T−1]).

By Proposition 10.12, the operation mbk is defined over this Q[[T 1/N ]][T−1].

We have thus obtained a filtered A∞ category over Q[[T 1/N ]][T−1], which wedenote by

FukN (M,ω).

To include all the BS-rational Lagrangian submanifolds and obtain a filtered

A∞ category over Λrat(0)nov we proceed as follows. Let L0, L1 be Lagrangian subman-

ifolds which are m0-BS rational and m1-BS rational, respectively. We take therem0 (resp. m1) rationalization SL0

(resp. SL0). Take any N such that m0, m1 | N .

SL0(resp. SL0

) induce an N rationalization Let SNL0

(resp. SNL1) in an obvious way.

(Namely SNL0

= (SL0)⊗N/m0 .)

51


For p ∈ L0 ∩ L1, we use (10.3) to obtain c(p). To make N -dependence of c(p)explicit, we write cN (p) for c(p). Then for each given (N,N ′) with N |N ′, we haveN ′cN (p)− cN ′(p) =: Δ(p) ∈ Z≥0. We put

(10.8) cN ′(p) = cN (p)− Δ(p)

N ′ .

We write [[p]]N and [[p]]N ′ to distinguish the generators of Floer chain complex over

Q[[T 1/N ]][T−1] and over Q[[T 1/N ′]][T−1]. We consider the map

[[p]]N �→ T−1/Δ(p)[[p]]N ′ ,

that induces an isomorphism

C(L1, L0;Q[[T 1/N ]][T−1])⊗Q[[T 1/N ]] Q[[T 1/N ′]][T−1]

−→ C(L1, L0;Q[[T 1/N ′]][T−1])

which respect all the A∞ operations.Therefore the system (FukN (M,ω);<) with respect to the partial order ‘N <

N ′ if and only if N | N ′’ forms an inductive system. We define the A∞-categoryFukrat(M,ω) to be the associated inductive limit.

Now let Z be the profinite completion of Z. As in [Fu3], we will define an action

of Z on Fukrat(M,ω). To define a Z action we need to include a flat line bundleL over L and take R = C in place of R = Q. Namely we take (L,L, sp, b, s, SL)where (L, sp, b, s, SL) is as before and L is a flat U(1) bundle over L. We say L isN -rational if the image of the holonomy representation π1(L) → U(1) is containedin {exp(2π

√−1k/N) | k ∈ Z}.

Now let (Li,Li, spi, bi, si, SLi) be as above such that Li are N -rational. We

put

(10.9)

C((L1,L1),(L0,L0);C[[T1/N ]][T−1])

=⊕

p∈L0∩L1

Q[[T 1/N ]][T−1][[p]]⊗Q HomC((L1)p, (L0)p).

We then modify operations mk by using the holonomy of Li: Namely we incorporatethe holonomy weight in U(1) as defined in (3.28) [Fu2] into the right hand side of(10.7). Taking an inductive limit in the same way, we obtain a filtered A∞ category

over Λrat (0)Cnov = Λ

rat (0)nov ⊗Q C.

The mambZ action on it is defined as follows: Let mamb ∈ mambZ/(NZ) be thestandard generator. We define an action on the set of object by

mamb · (L,L, sp, b, s, SL) = (L,L ⊗ PL, sp, b, s, SL)

Since P⊗N/mamb is a trivial bundle on L, this defines an action of mambZ/NZ.In the same way as [Fu3] this induces an action of mambZ/NZ on the category

over C[[T 1/N ]][T−1].

Remark 10.13. Note the (Galois) action of 1 on C[[T 1/N ]][T−1] is

T 1/N �→ exp (2π√−1/N)T 1/N .

This action is consistent with the above action, as was shown in [Fu3].

We then take the inductive limit and obtain an action ofmambZ on the category

over Λrat (0)Cnov . The proof of Theorem 1.2 is complete. �

52


Remark 10.14. We may take the maximal abelian extension of Q, that is thefield adding all the roots of unity to Q, in place of C.

Remark 10.15. We use the coefficient ring Λrat (0)Cnov which is the subring of

ΛratCnov consisting of the series not involving the grading parameter e. This is because

we include grading s in the object of our category and so the Floer cohomology hasabsolute Z grading such that all the operations mk is of degree 1 (after shifted).(We also choose the bounding cochain b so that it is degree 1.)

We may consider the ZN -grading instead, then the category is defined over

Λrat (0)Cnov [eN , e−N ].

If we take (L,L, sp, b, SL) as an object (that is we do not include grading atall) then the category is defined over ΛratC

nov

We note that the Z action exists in all of these versions.

10.3. The reduction of coefficient ring: anchored version. To see therelation between the construction of the last subsection to the critical value, it is use-ful to consider anchor. Let y be the base point of M we also fix Vy ∈ Lag(TyM,ω).We take and fix an element Sy ∈ Vy such that ‖Sy‖ = 1. Let (Li, γi) be anchoredLagrangian submanifolds. We assume Li are N -rational. Then it is easy to seethat there exists a unique N -rationalization SN such that SN (γi(1)) is a paralleltransport of Sy along γi. Using this rationalization we discuss in the same way asthe last subsection to obtain a non-anchored version: More specifically, we have thefollowing anchored version of Proposition 10.8.

Proposition 10.16. Let E be an anchored Lagrangian chain of length ≥ 2.Define a map E′

k : πad2 (E ; �p) → R

(10.10) E′k(B) :=

∫B

ω −k∑

i=0

c(p(i+1)i)

for k ≥ 2, and

E′1(α) =

∫α

ω − c(p10)

for k = 1 and α ∈ π2(�01; p01). Then E′k have their values in Z[1/N ] for all

k = 1, · · · and the collection E′ = {E′�} defines an abstract index on the collection

of N-rational anchored Lagrangian submanifolds.

The module of morphisms is

CF ((L1, γ1), sp, b1, λ1), ((L0, γ0), sp, b0, λ0))⊗Λ(L0,L1;�01) Λratnov

and higher products are defined in the same way as before. To define a Z action onthe corresponding A∞ category, we include flat U(1) bundles with finite holonomyon Li as before.

We remark that we obtain the same filtered A∞ category as the non-anchoredversion when M is simply connected. However in general the two are different.

References

[A] Arnold, V., On a characteristic class entering in quantization conditions, Funct. Anal.Appl. 1 (1967), 1-14.

[Fl1] Floer, A., Morse theory for Lagrangian interesections, J. Differ. Geom. 28 (1988),513–547.

53


[Fl2] Floer, A.,A relative Morse index for the symplectic action, Comm. Pure Appl. Math.41 (1988), 393–407.

[Fl3] Floer, A., The unregularized gradient flow of the symplectic action, Comm. Pure Appl.Math. 41 (1988), 775–813.

[Fu1] Fukaya, K., Morse homotopy, A∞-category, and Floer homologies, Proceedings ofGARC Workshop on Geometry and Topology ’93, 1–102, Lecture Notes Ser., 18,Seoul Nat. Univ., Seoul, (1993).

[Fu2] Fukaya, K., Floer homology and mirror symmetry II, Adv. Stud. Pure Math., 34,31–127, Math. Soc. Japan, Tokyo, (2002).

[Fu3] Fukaya, K., Galois symmetry on Floer cohomology, Turkish J. Math. 27, 11–32 (2003).[FOOO00] Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K., Lagrangian Intersection Floer Theory -

anomaly and obstruction, preprint 2000.[FOOO09] Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K., Lagrangian Intersection Floer Theory

- Anomaly and Obstruction, vol. 46 I & II, AMS/IP Studies in Advanced Math.,AMS/International Press, 2009.

[FOn] Fukaya, K., Ono, K., Arnold’s conjecture and Gromov-Witten invariants, Topology38 (1999), 933-1048.

[HS] Hofer, H., and Salamon, D., Floer homology and Novikov ring, in “The Floer MemorialVolume” Progr. Math. 133, ed. by H. Hofer, C. Taubes, A. Weinstein and E. Zehnder,Birkhauser, Basel (1995) 483–524

[KS] Kashiwara, M., and Schapira, J.P., Sheaves on Manifolds, Grundlehren der Mathema-tischen Wissenschaften 292 Springer-Verlag, Berlin, 1990.

[Ko] Kontsevich, M., Homological algebra of mirror symmetry, Proceedings of the Interna-tional Congress of Mathematicians, Vol. 1, 2 (Zuich, 1994), 120–139, Birkhaser, Basel,1995.

[N] Novikov, S., Multivalued functions and functional - an analogue of the Morse theory,Sov. Math. Dokl. 24 (1981) 222–225

[Oh1] Oh, Y.-G. Mean curvature vector and symplectic topology of Lagrangian submanifoldsin Einstein-Kahler manifolds, Math. Z. 216 (1994), 471 – 482.

[Oh2] Oh, Y.-G. Construction of spectral invariants of Hamiltonian paths on closed symplec-

tic manifolds, in ‘The Breadth of Symplectic and Poisson Geometry’, 525–570, Progr.Math., 232, Birkhauser Boston, Boston, MA, 2005.

[Oh3] Oh, Y.-G., Seidel’s long exact sequence on Calabi-Yau manifolds, preprint 2010,arXiv:1002.1648.

[On] Ono, K., On the Arnol’d conjecture for weakly monotone symplectic manifolds, Invent.Math. 119 (1995) 519–537

[Se1] Seidel, P., Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), 103–149.

[Se2] Seidel, P., A long exact sequence for symplectic Floer cohomology, Topology 42 (2003),1003 – 1063.

[V] Viterbo C. , Intersection des sou-varietes lagrangians, fonctionelles d’action et indicesdes system Hamiltoniennes, Bull. Soc. Math. France 115 (1987), 361–390.

[W] Welschinger, J.-Y., Open strings, Lagrangian conductors and Floer functor, preprint,arXiv:0812.0276.

Department of Mathematics, Kyoto University, Kyoto, Japan


Department of Mathematics, University of Wisconsin, Madison, WI, USA


Graduate School of Mathematics, Nagoya University, Nagoya, Japan


Department of Mathematics, Hokkaido University, Sapporo, Japan


54


Motivic Donaldson-Thomas invariants: summary of results

Maxim Kontsevich and Yan Soibelman

To the memory of I.M. Gelfand

Abstract. This paper is a digest of our longer paper [KoSo1] where the con-cept of motivic Donaldson-Thomas invariants for 3-dimensional Calabi-Yaucategory (3CY category for short) was introduced.

Contents

1. Introduction

This paper is a “digest” of our longer paper [KoSo1] where the concept ofmotivic Donaldson-Thomas invariants for 3-dimensional Calabi-Yau category (3CYcategory for short) was introduced.

From the point of view of a physicist, in [KoSo1] we offered a mathematicaldefinition of the notion of BPS invariants (including “refined BPS invariants”, cf.[GuDi]) in a very general, model independent, framework. In loc. cit. we alsoproved a general wall-crossing formula, which agrees with known wall-crossing for-mulas for N = 2 theories, e.g. with the one for SU(2) Seiberg-Witten theory, withDenef-Moore semiprimitive wall-crossing formula (see [DeMo]) and with Cecotti-Vafa work [CeVa] on soliton counting in 2d sypersymmetric QFTs.

From the point of view of a mathematician, we suggested a framework for thecounting of “motivic invariants” (e.g. Euler characteristic, Poincare polynomial,etc.) of 3CY categories (e.g. derived category of coherent sheaves on a compact orlocal 3-dimensional Calabi-Yau manifold).

Since the paper [KoSo1] is very involved and contains many constructions,ramifications and connections to many different topics, its logic can be hidden“behind the trees”. In this paper we are going to recall the general philosophy of[KoSo1] and discuss few applications and open problems.

Second author was partially supported by NSF FRG grant DMS-0854989. He also thanks toIHES for excellent research conditions.

c©0000 (copyright holder)

1



55

2 MAXIM KONTSEVICH AND YAN SOIBELMAN

Let us summarize the approach of [KoSo1] which leads in the end to the defi-nition of motivic and numerical Donaldson-Thomas (DT for short) invariants. Forthe motivations we refer to the Introduction in [KoSo1].

1. With an ind-constructible triangulated A∞-category C over a field k (see[KoSo1], Section 3) we associated (see [KoSo1], Section 6.1) a unital associativealgebra H(C) which is called motivic Hall algebra. Roughly speaking it is an alge-bra formed by motivic stack functions (see [Jo3] and [KoSo1], Section 4.2) on thespace (stack) of objects Ob(C). The structure constants of H(C) are defined via anappropriate count of exact triangles in the triangulated category C.

2. A choice of constructible stability condition on C (see [KoSo1], Section 3.4)allows us to define a collection of full subcategories CV ⊂ C parametrized by strictsectors V ⊂ R2. Each subcategory CV is generated by extensions of semistableobjects of C with central charges in V and by the zero object. More precisely, thedefinition of the stability condition depends on some data. Part of the data is ahomomorphism cl : K0(C) → Γ, where Γ is a free abelian group endowed withan integer-valued bilinear form. Another piece of the data is the central charge,i.e. homomorphism Z : Γ → C. A choice of V allows us to define a convex coneC(V, Z) ⊂ Γ⊗R, generated by some lattice elements γ for which Z(γ) ∈ V . Thenone can complete H(CV ) with respect to the cone and obtain a completed motivic

Hall algebra H(CV ).3. We define a collection of invertible elements AHall

V ∈ H(CV ), roughly, ascharacteristic functions of the stack of objects of CV (see [KoSo1], Section 6.1). Inthe case when k is a finite field one has

AHallV = 1 + · · · =

∑[E]∈Iso(CV )

[E]

#Aut(E).

Elements AHallV for generic V satisfy the following Factorization Property:

AHallV = AHall

V1·AHall

V2

for a clockwise decomposition of the strict sector into subsectors: V = V1�V2. Strictsectors are not required to be non-degenerate, open or closed, thus an example ofthe decomposition is a sector V dissected into two by a ray. Furthermore, theFactorization Property implies a factorization formula

AHallV =

−→∏l⊂V

AHalll

taken in the clockwise order over all rays l belonging to V .4. In case when C is an ind-constructible 3CY category over a field of char-

acteristic zero (see [KoSo1], Section 3.3) we define a motivic quantum torus RC ,which is an associative algebra described by the usual relations

eγ1eγ2

= L12 〈γ1,γ2〉eγ1+γ2

, e0 = 1 .

Here L12 is the square root of the motive L of the affine line [A1

k], elements γ belongto the lattice Γ, and we assume that the integer-valued bilinear form 〈γ1, γ2〉 is skew-symmetric. The coefficient ring for the quantum torus can be any commutative ringC. We use two choices for C: either a certain localization of the ring of motivicfunctions on Spec(k) or its l-adic version (more precisely we pass to rings formedby certain equivalence classes, see [KoSo1], Sections 4.5, 6.2).

56

MOTIVIC DONALDSON-THOMAS INVARIANTS: SUMMARY OF RESULTS 3

5. After a choice of the so-called orientation data (roughly, a square root of asuper line bundle sdetExt•(E,E) on the space of objects Ob(C), see [KoSo1], Sec-tion 5) we define motivic Donaldson-Thomas invariants as a collection of elementsAmot

V ∈ RCVwhich are images of AHall

V under the homomorphisms

ΦV : H(CV ) → RCV

defined in Section 6.3 of [KoSo1]. Here we slightly abuse the notation denoting byRCV

a completion of RC which depends on a choice of V (see Section 4.2. below).More precisely, we proved the existence of the collection of homomorphisms ΦV

in the l-adic case which is sufficient for applications to numerical DT-invariants. Thegeneral motivic case depends on a conjectural integral formula (Conjecture 4 statedin Section 4.4 of [KoSo1]). The definition of the homomorphism ΦV relies on thetheory ofmotivic Milnor fiberMF (WE) of the potentialWE of an object E ∈ Ob(C).The potential is defined in terms of the Calabi-Yau structure. The motivic Milnorfiber satisfies the motivic Thom-Sebastiani theorem proved by Denef and Loeser(see [DenLo]). The latter is used in the proof that ΦV is a homomorphism of rings.

Since the elements AHallV satisfy the Factorization Property, the same is true

for the elements AmotV .

6. Having the motivic DT-invariant (which can be informally thought of asSerre polynomial associated with the mixed Hodge module of vanishing cycles onOb(C) of the potential WE) we can obtain numerical DT-invariants by a proceduresimilar to passing from the Poincare polynomial to the Euler characteristic, i.e.a kind of “quasi-classical limit”. The situation is complicated by the fact thatthe elements Amot

V defined above do not have such a limit. We conjectured in[KoSo1], Section 7.3. that there exists a specialization of the automorphism x →Amot

V x(AmotV )−1 of the (completed) motivic quantum torus RCV

at L12 = −1. It

gives a formal Poisson automorphism AV of the Poisson torus, whose algebra offunctions is generated by the elements eγ which are quasi-classical limits of eγ .

7. We define birational Poisson automorphisms of the Poisson torus associatedwith Γ such as follows:

Tγ : eμ → (1− eγ)〈γ,μ〉eμ, γ, μ ∈ Γ .

Then each AV belongs to the pronilpotent group of formal Poisson automorphismsof the above torus and is given by the formula:

AV =−→∏

γ∈C(V,Z)∩Γ

T Ω(γ)γ

where Ω(γ) are rational numbers, Z is the central charge, and the product is takenin the clockwise direction with respect to the arguments of Z(γ) (more precisely onedefines first a factor corresponding to a single ray l = R>0Z(γ), and then combinesthese factors into the clockwise product, see the details in the loc. cit). Theseare numerical DT-invariants of the category C (BPS invariants in the languageof physics). We conjectured in Section 7.1 of [KoSo1] that Ω(γ) are integers fora generic stability condition, and we proved the integrality in [KoSo3] for quiverswith potential. Also, conjecturally they do not depend on the orientation data.

The above definition of Ω(γ) agrees with the Behrend local invariant

(−1)dimX(1− χ(MFE(WE))) ,

57


(see [Be1]) where in the RHS we compute the Euler characteristic of the Milnorfiber of the potential WE (informally, the set of objects of C belongs to the set ofcritical points of WE considered as a function on a larger space X). More precisely,the invariants agree for the category consisting of Schur objects. For that reasonour motivic DT-invariants can be thought of as a quantization of DT-invariantsintroduced by Behrend in [Be1].

Summarizing, the definition of motivic DT-invariants (refined BPS invariantsin the language of physics) of the category C depends a priori on two conjectures:existence of orientation data and the the integral identity conjecture from Section4.4 of [KoSo1]. The definition of the numerical DT-invariants Ω(γ) depends on theconjecture about existence of the quasi-classical limit of motivic DT-invariants (seeparagraph 6 above). It does not depend on the full version of the integral identity,since we can use the l-adic version of the theory (including e.g. the l-adic versionof the quantum torus), for which the integral identity was proved in Section 4.4of [KoSo1]. Also, we expect that numerical DT-invariants do not depend on theorientation data. As we already pointed out, our motivic DT-invariants can beinformally thought of as Hilbert series of the equivariant cohomology of the motivicperverse sheaf of vanishing cycles of the potential on the stack of objects of our3CY category. In some cases (e.g. for the Calabi-Yau category associated withquiver with potential) we can give more precise meaning to the above phrase. Moreon that can be found in our paper [KoSo3], where we offer an alternative (andmore elementary) point of view on motivic DT-invariants. Although in [KoSo3]we consider only 3CY categories associated with representations of a quiver withpotential, it seems that the approach we propose there is more general. More pre-cisely, we expect that an ind-constructible locally regular 3CY category (satisfyingsome additional conditions) can be locally described by a quiver with potential.Then the Hilbert series from [KoSo3] (see also Section 7.3 below) is the motivicDT-invariant.

About the structure of the paper. It consists of two parts. In the first (larger)part we review the constructions from [KoSo1] which lead to motivic and numericalDT-invariants. Most of the material is borrowed from [KoSo1]. In the second partwe discuss several speculations and conjectures from [KoSo1]. Some of them deservea special project. In particular, as our original motivation for [KoSo1] was our paper[KoSo4], we stress here again that a good way to encode our DT-invariants is asgluing data for a complex integrable system (or its quantization) “a’la [KoSo4]”.The DT-invariants are used for the definition of the gluing automorphisms. Wall-crossing formulas ensure that the gluing is well-defined. Maybe this integrablesystem (or rather its quaternion Kahler cousin, see [KoSo5]) is the main structureunderlying the mathematical theory of BPS states.

We should say that we have not attempted to mention here contributions ofall researchers working in the area of Donaldson-Thomas invariants and their gen-eralizations. As a result our list of references contains very few papers. We donot discuss here the motivations and the history of the subject, just mentioningthat [KoSo1] was mostly motivated by our previous work [KoSo4]. At the sametime, there are several papers on DT-invariants written by other authors which arelogically related to [KoSo1]. We would like to mention just few of them for the sakeof the reader.

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1. The approach to DT-invariants via critical points of a function is due toBehrend (see [Be1], [BeFa]). Our paper [KoSo1] can be thought of as a “quantiza-tion” of [Be1], with the quantization parameter being the motive of affine line.

2. Joyce was probably first who approached “DT-type” invariants in abstractcategorical setting. He developed the technique of motivic stack functions and un-derstood the relevance of motives to the counting problem (see [Jo1-Jo4]). Themain limitation of his approach was due to the fact that he did not use the notionof potential and worked with abelian rather than triangulated categories (for manyapplications, especially to physics, one needs triangulated categories). His recenttheory of “generalized DT-invariants” developed jointly with Song (see [JoSo1,2])fixes some of these gaps and fits well with the general philosophy of [KoSo1] (infact the papers [JoSo1,2] use some parts of [KoSo1]). Those papers as well as thepapers by Pandharipande and Thomas (see [PT1,2]) deal with concrete examplesof categories (e.g. the category of coherent sheaves). The authors construct numer-ical invariants via Behrend approach. It is difficult to prove that they are in factinvariants of triangulated categories (which is manifest in [KoSo1]).

3. The general concept of BPS state when encoded mathematically uses thenotion of semistable object in a triangulated category. Hence the theory of stabil-ity conditions of Bridgeland (see [Br1]) is a necessary tool. Motivated by [Jo1-4]Bridgeland developed (jointly with Toledano Laredo) an approach to Joyce invari-ants based on the theory of Stokes factors for irregular connections on P1 (see[BrTL]). As we explained in Section 2.7 of [KoSo1] this is a special case of ourgeneral theory of stability data on graded Lie algebras. This part of the story isindependent on the categorical framework, and we will only briefly discuss it belowin Section 5 (for more details see [KoSo1], Section 2).

Acknowledgements. Second author was partially supported by NSF FRG grantDMS-0854989. He also thanks to IHES for excellent research conditions.

2. Ind-constructible 3CY -categories

For an introduction to the geometric language of A∞-categories and axiomaticsof triangulated A∞-categories we refer the reader to [KoSo2] and the Appendix of[So2]. For a more algebraic exposition of A∞-categories the reader can look at [Ke1].In any case, we will assume the familiarity with some basic notions. For instancethe reader should know that for an A∞-category there are infinitely many highercomposition maps mn, n ≥ 1, such that m1 : Hom(E1, E2) → Hom(E1, E2)[1]is a differential and the composition map m2 : Hom(E1, E2) ⊗ Hom(E2, E3) →Hom(E1, E3) is associative only on the cohomology groups with respect to m1.Differential-graded categories are examples ofA∞-categories with allmn = 0, n ≥ 3.

For simplicity we will assume that all A∞-categories are defined over the groundfield k of characteristic zero (although this condition can be relaxed for some results,e.g. for the definition of motivic Hall algebra).

2.1. Ind-constructible triangulated A∞-categories, Calabi-Yau cate-gories, stability conditions. Recall that a Calabi-Yau category of dimension dis a weakly unital k-linear triangulated A∞-category such that for any two ob-jects E,F the Z-graded vector space Hom•(E,F ) = ⊕n∈Z Homn(E,F ) is finite-dimensional and moreover:

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1) We are given a non-degenerate pairing

(•, •) : Hom•(E,F )⊗Hom•(F,E) → k[−d] ,

which is symmetric with respect to interchange of objects E and F ;2) For any N � 2 and a sequence of objects E1, E2, . . . , EN we are given a

polylinear Z/NZ-invariant map

WN : ⊗1≤i≤N (Hom•(Ei, Ei+1)[1]) → k[3− d] .

Here [1] means the shift in the category of Z-graded vector spaces, and we setEN+1 = E1;

Explicitly, the maps WN are defined in terms of the higher compositions mn

and the pairing by the formula

WN (a1, . . . , aN ) = (mN−1(a1, . . . , aN−1), aN ) .

The collection (WN )N�2 is called the potential of C. If d = 3 then for any objectE ∈ Ob(C) we define a formal series W tot

E at 0 ∈ Hom•(E,E)[1] by the formula:

W totE (α) =

∑n�2

Wn(α, . . . , α)

n.

We call W totE the total (or full) potential of the object E. We call the potential

of E the restriction of W totE to the subspace Hom1(E,E). We will denote it by

WE . One can prove (see [KoSo1], Proposition 7) that the potential WE admits adecomposition

WE = WminE ⊕QE ⊕NE ,

where WminE is the potential of the minimal model Cmin (i.e. it is a formal series on

Ext (1 E,E)), the quadratic formQE is defined on vector space Hom (1 E,E)/Ker(m1:

Hom1(E,E) → Hom2(E,E)) by the formula QE(α, α) = m2(α,α)2 , and NE is the

zero function on the image of the map m1 : Hom0(E,E) → Hom1(E,E). In theabove splitting formula we use the notation (f⊕g)(x, y) = f(x)+g(y) for the directsum of formal functions f and g.

In order to apply the techniques of the theory of motivic integration we con-sider a class of A∞-categories which we call ind-constructible. Recall the followingdefinition.

Definition 1. Let S be a variety over k. A subset X ⊂ S(k) is called con-structible over k if it belongs to the Boolean algebra generated by k-points of open(equivalently closed) subschemes of S.

Ind-constructible sets are colimits of constructibles ones (equivalently, count-able unions of non-intersecting constructible sets). They naturally form a symmet-ric monoidal category. The notion of ind-constructible A∞-category developed in[KoSo1], Section 3.1 as well as its Calabi-Yau version developed in loc.cit. Sec-tion 3.3, roughly speaking, means that objects of such a category form an ind-constructible set, morphisms form constructible bundles, higher composition mapsare morphisms of tensor products of such bundles, etc. We impose the conditionthat near each object all the structures are in fact regular i.e. described by schemes,regular maps, etc. In particular, the potential WE can be thought of as a “partiallyformal” function: it is regular along Ob(C) near E ∈ Ob(C) and formal in the di-rection α ∈ Hom1(E,E). We skip here many delicate details, referring the readerto Section 3 of [KoSo1].

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Remark 1. In Section 3.2 of [KoSo1] we explain how to associate to an ind-constructible category C over k an ordinary k-linear A∞-category C(k). This con-struction is not quite trivial, since the isomorphism classes of objects in C in theconstructible setting depend only on the naturally defined category C(k) over thealgebraic closure k of k. The correct definition involves descent data associatedwith all Galois extensions of k. Because of that we impose the restriction that theground field k is perfect.

The class of ind-constructible A∞-categories is large due to Bondal-van denBergh theorem (see [BonVdB]), which makes many geometrically defined categories(e.g. Db(X) for a smooth projective scheme X) to be equivalent to the categoryA-mod of A∞-modules over an A∞-algebra with finite-dimensional cohomology. Inthis “derived non-commutative geometry” sense all schemes become affine. Thusone can describe objects of a category by an increasing number of solutions topolynomial equations solved in the free modules An. This implies that the objectsform and ind-constructible set. Similarly one can treat morphisms.

A modification of Bridgeland theory of stability conditions to the case of ind-constructible triangulated categories was suggested in Section 3.4 of [KoSo1]. Letus briefly recall it here.

Let C be an ind-constructible weakly unital A∞-category over a field k of ar-bitrary characteristic. Let cl : Ob(C) → Γ � Zn be a map of ind-constructible sets(where Γ is considered as a countable set of points) such that the induced mapOb(C)(k) → Γ factorizes through a group homomorphism clk : K0(C(k)) → Γ.It is easy to see that for any field extension k′ ⊃ k we obtain a homomorphism

clk′ : K0(C(k′)) → Γ.

Definition 2. A constructible stability structure on (C, cl) is given by the fol-lowing data:

• an ind-constructible subset

Css ⊂ Ob(C)consisting of objects called semistable, and satisfying the condition thatwith each object it contains all isomorphic objects,

• an additive map Z : Γ → C called central charge, such that Z(E) :=Z(cl(E)) �= 0 if E ∈ Css,

• a choice of a branch of the logarithm LogZ(E) ∈ C for any E ∈ Css whichis constructible as a function of E.

These data are required to satisfy the usual Bridgeland axioms (see [Br1]). Inparticular, for any object we have its Harder-Narasimhan filtration. We imposealso the following new axioms:

• the set of E ∈ Css(k) ⊂ Ob(C)(k) with fixed cl(E) ∈ Γ \ {0} and fixedLogZ(E) is a constructible set.

• (Support Property) Pick a norm ‖ · ‖ on Γ⊗R. Then there exists C > 0such that for all E ∈ Css one has ‖ E ‖≤ C|Z(cl(E))|.

Slightly ahead of our exposition we remark here that the Support Property isequivalent to the one for stability data on graded Lie algebras, which we are goingto use below in Section 5.1 (see there for the details):

There exists a quadratic form Q on ΓR := Γ⊗R such that1) Q|Ker Z < 0;

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2) Supp a ⊂ {γ ∈ Γ \ {0}| Q(γ) � 0},where we use the same notation Z for the natural extension of Z to ΓR. We

are going to explain in Section 5 that the notion of stability data a = (a(γ))γ∈Γ ona graded Lie algebra leads naturally to symplectomorphisms generalizing AV .

2.2. Motivic stack functions. The theory of motivic functions introducedby the first author was developed by many people, most notably, by Denef andLoeser. Its equivariant version (theory of motivic stack functions) was developedindependently by Joyce (see [Jo3]). We recall below a version based on Sections4.1, 4.2 of [KoSo1].

The abelian group Mot(X) of motivic functions is the group generated bysymbols [π : S → X] := [S → X] where π is a morphism of constructible sets,subject to the relations

[(S1 � S2) → X] = [S1 → X] + [S2 → X] .

For any constructible morphism f : X → Y we have two homomorphisms of groups:1) f! : Mot(X) → Mot(Y ), defined by [π : S → X] → [f ◦ π : S → X];2) f∗ : Mot(Y ) → Mot(X), defined by [S′ → Y ] → [S′ ×Y X → X].Moreover, Mot(X) is a commutative ring via the fiber product operation. We

denote by L ∈ Mot(Spec(k)) the element [A1k] := [A1

k → Spec(k)]. It is customaryto add its formal inverse L−1 to the ring Mot(Spec(k)) (or more generally to thering Mot(X) which is a Mot(Spec(k))-algebra).

Let us recall several “realizations” of motivic functions. For each realizationthe theory of motivic DT-invariants developed in [KoSo1] has the correspondingversion (except, maybe, the case (iv) when the notion of potential does not makesense).

(i) There is a homomorphism of rings

χ : Mot(X) → Constr(X,Z) ,

where Constr(X,Z) is the ring of integer-valued constructible functions on X en-dowed with the pointwise multiplication. More precisely, the element [π : Y → X]is mapped into χ(π), where χ(π)(x) = χ(π−1(x)), which is the Euler characteristicof the fiber π−1(x).

(ii) Let now X be a scheme of finite type over a field k, and l �= char k be aprime number. There is a homomorphism of rings

Mot(X) → K0(Dbconstr(X,Ql)) ,

where Dbconstr(X,Ql) is the bounded derived category of etale l-adic sheaves on X

with constructible cohomology. It is defined by the formula

[π : S → X] → π!(Ql) ,

which is the direct image in the derived sense of the constant sheaf Ql. Notice thatDb

constr(X,Ql) is a tensor category, hence the Grothendieck group K0 is naturallya ring. In this context the homomorphisms f! and f∗ discussed above correspondto the functors f! (direct image with compact support) and f∗ (pullback), whichwe will denote by the same symbols. We will also use the notation

∫Xφ := f! (φ)

for the canonical map f : X → Spec(k).(iii) In the special case X = Spec(k) the above homomorphism becomes a map

[S] →∑i

(−1)i[Hic(S ×Spec(k) Spec(k),Ql)] ∈ K0(Gal(k/k)−modQl

) ,

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where Gal(k/k)−modQlis the tensor category of finite-dimensional continuous l-

adic representations of the Galois group Gal(k/k), and we take the etale cohomologyof S with compact support.

(iv) If k = Fq is a finite field then for any n � 1 we have a homomorphism

Mot(X) → ZX(Fqn )

given by[π : Y → X] → (x → #{y ∈ X(Fqn) |π(y) = x}) .

Here the operations f !, f∗ correspond to pullbacks and pushforwards of functionson finite sets.

(v) If k ⊂ C then the category of l-adic constructible sheaves on a scheme offinite type X can be replaced in the above considerations by Saito’s category ofmixed Hodge modules.

(vi) In the case X = Spec(k) one has two additional homomorphisms:a) The Serre polynomial

Mot(Spec(k)) → Z[q1/2]

defined by

[Y ] →∑i

(−1)i∑

w∈Z�0

dimHi,wc (Y )qw/2 ,

where Hi,wc (Y ) is the weight w component in the i-th Weil cohomology group with

compact support.b) If char k = 0 then we also have the Hodge polynomial

Mot(Spec(k)) → Z[z1, z2]

given by

[Y ] →∑i�0

(−1)i∑

p, q�0

dimGrpF (GrWp+q HiDR,c(Y ))zp1z

q2 ,

where GrW• and Gr•F denote the graded components with respect to the weightand Hodge filtrations, and Hi

DR,c denotes the de Rham cohomology with compactsupport.

Clearly the Hodge polynomial determines the Serre polynomial via the homo-morphism Z[z1, z2] → Z[q1/2] such that zi → q1/2, i = 1, 2.

Let X be a constructible set over a field k and G be an affine algebraic groupacting on X, in the sense that G(k) acts on X(k) and there exists a G-variety Sover k with a constructible equivariant identification X(k) � S(k).

We define the groupMotG(X) of G-equivariant motivic functions as the abeliangroup generated by all G-equivariant constructible maps [Y → X] modulo therelations

• [(Y1 � Y2) → X] = [Y1 → X] + [Y2 → X],• [Y2 → X] = [(Y1 ×Ad

k) → X] if Y2 → Y1 is a G-equivariant constructiblevector bundle of rank d.

This group is a commutative ring via the fiber product, and a morphism of con-structible sets with group actions induces a pullback homomorphism of correspond-ing rings. There is no natural operation of a pushforward for equivariant motivicfunctions.

Let X be a constructible set acted by an affine algebraic group G. It defines anobject (X,G) in the 2-category of constructible stacks. The abelian group of stack

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motivic function Motst((X,G)) is generated by the group of isomorphism classesof 1-morphisms of stacks [(Y,H) → (X,G)] with the fixed target (X,G), subject tothe relations

• [((Y1, G1) � (Y2, G2)) → (X,G)] = [(Y1, G1) → (X,G)] + [(Y2, G2) →(X,G)]

• [(Y2, G1) → (X,G)] = [(Y1 × Adk, G1) → (X,G)] if Y2 → Y1 is a G1-

equivariant constructible vector bundle of rank d.

The ring MotG(X) maps to Motst((X,G)).Finally, for a constructible stack S = (X,G) we define its class in the ring

K0(V ark)[[L]−1, ([GL(n)]−1)n�1] as

[S] = [(X ×GL(n))/G]

[GL(n)],

where we have chosen an embedding G → GL(n) for some n � 1, and (X ×GL(n))/G is the ordinary quotient by the diagonal free action (thus in the RHSwe have the quotient of motives of ordinary varieties). The result does not de-pend on the choice of embedding. Then we define the integral

∫S : Motst(S) →

K0(V ark)[[L]−1, ([GL(n)]−1)n�1] as

∫S [S ′ → S] = [S ′].

Furthermore, in Section 4.5 of [KoSo1] we associated with the ring of motivic

function Mot(X) (and with all equivariant versions of it) another ring Mot(X) ofcertain equivalence classes of motivic functions. The new ring has e.g. the propertythat two quadrics which have the same rank and determinant define in it the sameelement (this is probably not true for the usual motivic rings, see also the section

on orientation data below). In Section 4.6 of [KoSo1] we explained that Mot(X)and its equivariant cousin can be realized in terms of the rings of functions withnumerical (rather than motivic) values. This property allows us to prove identitiesin motivic rings by looking at the reduction of the corresponding schemes mod pwhere p is a sufficiently large prime number.

2.3. The motivic Milnor fiber. We collect here some facts from [KoSo1],Section 4.3. Let M be a complex manifold, x0 ∈ M . Recall, that for a germf of an analytic function at x0 such that f(x0) = 0 one can define its Milnorfiber MFx0

(f), which is a locally trivial C∞-bundle over S1 of manifolds with theboundary (defined only up to a diffeomorphism):

{z ∈ M | dist(z, x0) ≤ ε1, |f(z)| = ε2} → S1 = R/2πZ ,

where z → Arg f(z). Here dist is any smooth metric on M near x0, and thereexists a constant C = C(f, dist) and a positive integer N = N(f) such that for allchoices 0 < ε1 ≤ C and 0 < ε2 < εN1 the C∞ type of the bundle is the same for allε1, ε2, dist.

In particular, taking the cohomology of the fibers of MFx0(f) we obtain a

well-defined local system on S1.There are several algebro-geometric versions of this construction (theories of

nearby cycles). They produce analogs of the local system on S1, for example l-adicrepresentations of the group Gal(k((t))sep/k((t))) where l �= char k. In [KoSo1] weused two motivic versions of this notion: one developed by Denef and Loeser andanother one (in the framework of non-archimedean analytic geometry) developedby Nicaise and Sebag. In both cases the authors assume that char k = 0. Let Xbe a scheme over k. We introduce the group μ = lim←−n

μn and assume that μ acts

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trivially on X (here μn is the group of n-th roots of 1 in k). Then we have the groupMotμ(X). We will assume that the μ-action is “good” in the sense that μ acts viaa finite quotient μn and every orbit is contained in an affine open subscheme.

Let M be a smooth formal scheme over k with closed point x0 and f be aformal function on M vanishing at x0 (e.g. M could be the formal completion at0 of a fiber of vector bundle V → X in the above notation). We assume that f isnot identically equal to zero near x0.

Let us choose a simple normal crossing resolution of singularities π : M ′ → Mof the hypersurface in M given by the equation f = 0 with exceptional divisorsDj , j ∈ J . This follows from Hironaka’s theorem about resolution of singularities.In fact we need a canonical resolution of singularities, see e.g. [BiMi], [Te1,2] whichbetter suits the equivariant framework we are dealing with. Alternatively, as weexplained in [KoSo1], one can use the definition of the motivic Milnor fiber based onBerkovich theory of non-archimedean analytic spaces (see [NiSe]). Both approacheswork in case when f is a formal series (e.g. in the case of the approach via canonicalresolution of singularities it follows from [Te3]). The explicit formula for the motivicMilnor fiber in terms of a resolution of singularities looks as follows

MFx0(f) =

∑I⊂J,I �=∅

(1− L)#I−1[DI0 ∩ π−1(x0)] ∈ Motμ(Spec(k)) ,

where DI = ∩j∈IDj , D0I is the complement in DI to the union of all other excep-

tional divisors, and DI0 → D0

I is a certain Galois cover with Galois group μmI,

where mI is the g.c.d. of the multiplicities of all divisors Di, i ∈ I. Informally

speaking, the fiber of the cover DI0 → D0

I is the set of connected components of anon-zero level set of the function f ◦ π near a point of D0

I .Furthermore, the group Motμ(X) carries a non-trivial associative commutative

product introduced by Looijenga (which we call “exotic product” in Section 4.3 of[KoSo1]). We define

Mμ(X) := (Motμ(X), exotic product) .

Let V → X, V ′ → Y be two constructible vector bundles endowed with con-structible families f, g of formal power series. We denote by f ⊕ g the sum ofpullbacks of f and g to the constructible vector bundle

pr∗XV ⊕ pr∗Y V′ → X × Y .

The main result on motivic Milnor fibers that we need is the following motivicThom-Sebastiani theorem proved by Denef and Loeser (see [DeLo]).

Theorem 1. One has

(1−MF (f ⊕ g)) = pr∗X(1−MF (f)) · pr∗Y (1−MF (g)) ∈ Mμ(X × Y ) .

The version of the motivic Milnor fiber defined by means of non-archimedeananalytic geometry in [NiSe] agrees with the formula of Denef and Loeser. Onthe other hand, in the non-archimedean approach one can mimick the classicalconstruction which we recalled at the beginning of this section. Also, the abovedefinitions can be extended to the equivariant setting as well as to the case when fdepends on parameters. For our purposes the function f will be the potential WE

considered in a neighborhood of an object.

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2.4. Orientation data. The definition of motivic DT-invariants depends onthe minimal model of the potential. This means that we make a non-commutativechange of formal variables so that the Taylor series of the potential WE starts withthe terms of degree at least 3. This property is not invariant under the change ofan object: quadratic terms can appear if we decompose the same series at a nearbyobject. It turns out that the consistent choice of the quadratic part depends onan additional structure, which we call orientation data in [KoSo1], Section 5. Werecall it here.

Let C be an ind-constructible k-linear 3-dimensional Calabi-Yau category. Thenwe have a natural ind-constructible super line bundle D over Ob(C) with the fiberover E given by DE = sdet(Ext•(E,E)). It follows that on the ind-constructiblestack of exact triangles E1 → E2 → E3 we have an isomorphism of the pull-backedline bundles which fiberwise reads as

DE2⊗D−1

E1⊗D−1

E3� (sdet(Ext•(E1, E3)))

⊗2 .

Let Ob(C) = �i∈IYi be a decomposition into a disjoint countable union ofconstructible sets, each acted by an affine algebraic group GL(Ni).

Definition 3. Orientation data on C consists of a choice of an ind-constructiblesuper line bundle

√D on Ob(C) such that its restriction to each Yi, i ∈ I is GL(Ni)-

equivariant, endowed on each Yi with GL(Ni)-equivariant isomorphisms (√D)⊗2 �

D and such that for the natural pull-backs to the ind-constructible stack of exacttriangles E1 → E2 → E3 we are given equivariant isomorphisms:

√DE2

⊗ (√DE1

)−1 ⊗ (√DE3

)−1 � sdet(Ext•(E1, E3))

such that the induced equivariant isomorphism

DE2⊗D−1

E1⊗D−1

E3� (sdet(Ext•(E1, E3)))

⊗2

coincides with the one which we have a priori.

Orientation data are in a sense similar to a choice of spin structure on La-grangian submanifolds in the definition of Fukaya category (there are several othersituations where one has to make such a choice in order to define objects of a cat-egory). As we mentioned above, our story is related to the fact that the minimalmodel of the potential WE at an object E gives rise to a non-minimal potential WF

for a close object F . To say it differently, let V be a k-vector space endowed witha non-degenerate quadratic form Q. We define an element

I(Q) = (1−MF0(Q))L− 12 dimV ∈ Mμ(Spec(k))[L±1/2] ,

where L1/2 is a formal symbol which satisfies the relation (L1/2)2 = L, and Q is

interpreted as a function on V . Then the motivic Thom-Sebastiani theorem impliesthat

I(Q1 ⊕Q2) = I(Q1)I(Q2) .

Also we have I(Q) = 1, if Q is a split form: Q =∑

1≤i≤n xiyi for V = k2n. This

means that we can “twist” the element (1 − MF (W )) by I(Q). In all motivicrealizations (e.g. by taking the Serre polynomial) the element I(Q) depends onlyon the pair (rkQ, det(Q)mod(k×)2). It is not known (and probably not true) ifthe motive of a quadric is uniquely determined by the above pair of numbers. Thisproblem readily translates to the case of constructible families. As we will seebelow, the homomorphism from the motivic Hall algebra to the motivic quantum

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torus depends on the choice of quadratic part QE of the potential WE (or rather onthe corresponding element I(Q)). In order to make a consistent choice of the latterwe need to pick orientation data. The general problem of its existence is open (theanswer is positive for 3CY categories corresponding to quivers with potential).

3. Motivic Hall algebra

Let Ob(C) = �i∈IXi be a constructible decomposition of the space of objectswith the group GL(Ni) acting on Xi. Let us consider the Mot(Spec(k))-module⊕iMotst(Xi, GL(Ni)) and extend it by adding free generators which are negativepowers Ln, n < 0 of the motive of the affine line L. We denote the resultingMot(Spec(k))-module by H(C). We understand elements of H(C) as measures(and not as functions), because in the definition of the product we will use thepushforward maps. In the Section 6.1 of [KoSo1] we introduced a structure ofan asociative algebra on H(C) and called it motivic Hall algebra. The product isdefined on constructible families by the formula

[Y1 → Ob(C)] · [Y2 → Ob(C)] =∑n∈Z

[Wn → Ob(C)]L−n ,

where

Wn ={(y1, y2, α) | yi ∈ Yi, α ∈ Ext1(π2(y2), π1(y1)) , (π2(y2), π1(y1))≤0 = n

},

with the notation

(E,F )≤N :=∑i≤N

(−1)i dimExti(E,F ) .

The map Wn → Ob(C) is given by the formula

(y1, y2, α) → Cone(α : π2(y2)[−1] → π1(y1)) .

In other words, the structure constants are given by motives of exact triangles.For a constructible stability structure on C with an ind-constructible class mapcl : K0(C) → Γ, a central charge Z : Γ → C, a strict sector V ⊂ R2 and a branchLog of the logarithm function on V we have the category CV := CV,Log with objectswhich are either the zero object or an extension of semistable objects E such thatZ(cl(E)) ∈ V . Hence we have the associative algebra given by the completion

H(CV ) :=∏

γ∈(Γ∩C(V,Z,Q))∪{0}H(CV ∩ cl−1(γ)) ,

where C(V, Z,Q) is the convex cone spanned by elements γ �= 0 such that Z(γ) ∈ Vand Q(γ) ≥ 0 (recall here the quadratic form Q introduced at the end of Section2.1).

For every such algebra we define an invertible element AHallV ∈ H(CV ) such that

AHallV := 1 + · · · =

∑i∈I

1(Ob(CV )∩Yi,GL(Ni)) ,

where 1S is the identity function but interpreted as a counting measure. Then weproved the following easy result.

Proposition 1. The elements AHallV satisfy the Factorization Property:

AHallV = AHall

V1·AHall

V2

for a strict sector V = V1 � V2 (decomposition in the clockwise order).

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4. Quantum torus and motivic DT-invariants

4.1. Motivic weights. Let C be a 3-dimensional ind-constructible Calabi-Yaucategory over a field k of characteristic zero. We may assume that it is minimalon the diagonal, which means that we replaced every A∞-algebra End(E,E) by itsminimal model (i.e. m1 = 0). Then for any E ∈ Ob(C)(k) we have the potentialWmin

E which is a formal power series in α ∈ Ext1(E,E) which starts with cubicterms. It is just the minimal model of WE . We denote by

MF (E) := MF0(WminE )

the motivic Milnor fiber of WminE at 0 ∈ Ext1(E,E). Then the assignment E →

MF (E) can be interpreted as the value of some function MF ∈ Mμ(Ob(C)), whereμ, as before, is the group of all roots of 1.

Let us choose orientation data√D for C. This allows us to speak about a

super line bundle√D ⊗ D−1

≤1 with trivialized tensor square. Here D≤1 is a super

line bundle over the space of objects with the fiber at E ∈ Ob(C) given by

D≤1,E := sdet(τ≤1(Ext•(E,E))) ,

where τ≤i, i ∈ Z denotes the standard truncation functor.

The data of a super line bundle√D ⊗ D−1

≤1 is basically the same the data

consisting of a pair (V,Q) where V is a super vector space endowed with a quadratic

form Q. Thus we can define I(Q) = (1 − MF0(Q))L− 12 rkQ as an element of the

appropriate motivic ring.

Definition 4. The motivic weight w ∈ Mμ(Ob(C)) is the function defined onobjects by the formula

w(E) = L12

∑i≤1(−1)i dimExti(E,E)(1−MF (E))(1−MF0(QE))L

− 12 rkQE .

4.2. The motivic quantum torus. Let Γ be a free abelian group endowedwith a skew-symmetric, integer-valued bilinear form 〈•, •〉. For any commutativeunital ring C which contains an invertible symbol L1/2 we introduce a C-linearassociative algebra

RΓ,C := ⊕γ∈ΓC · eγwhere the generators eγ , γ ∈ Γ satisfy the relations

eγ1eγ2

= L12 〈γ1,γ2〉eγ1+γ2

, e0 = 1 .

We will call it the quantum torus associated with Γ and C.For any strict sector V ⊂ R2 we define

RV,C :=∏

γ∈Γ∩C0(V,Z,Q)

C · eγ

and call it the quantum torus associated with V . Here

C0(V, Z,Q) := C(V, Z,Q) ∪ {0}and C(V, Z,Q) is the convex cone generated by the set of vectors x ∈ ΓR :=Γ ⊗ R \ {0} such that Z(x) ∈ V and Q(x) ≥ 0. The algebra RV,C is the naturalcompletion of the subalgebra RV,C ∩ RΓ,C ⊂ RΓ,C . Notice that a choice of thesector V allows us to define the completion of RΓ,C which contains the motivicDT-invariant (see below). Being an infinite series the motivic DT-invariant doesnot belong to RΓ,C .

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Consider next the ring

Dμ = Mμ(Spec(k))[L−1,L1/2, ([GL(n)]−1)n�1] ,

where L = [A1k] is the motive of the affine line. The element L1/2 is a formal symbol

satisfying the equation (L1/2)2 = L. Instead of inverting the motives

[GL(n)] = (Ln − 1)(Ln − L) . . . (Ln − Ln−1)

of all general linear groups we can invert motives of all projective spaces

[Pn] =Ln+1 − 1

L− 1.

We also will consider the ringDμ of equivalence classes of functions inDμ mentionedin the section 2.2. The ring Dμ will play the role of the universal coefficient ringwhere motivic Donaldson-Thomas invariants take value. In particular, we choose itas the coefficient ring C in the above definition of the quantum torus. We denoteby RΓ := RΓ,Dμ the corresponding quantum torus and call it the motivic quantumtorus associated with Γ. Similarly, we have motivic quantum tori RV associatedwith strict sectors V ⊂ R2.

4.3. From motivic Hall algebra to motivic quantum torus. Assumethat C is an ind-constructible 3d Calabi-Yau category over a field k of characteristiczero, endowed with a constructible stability condition and orientation data

√D.

The Hall algebra of C is graded by the corresponding lattice Γ: H(C) = ⊕γ∈ΓH(C)γ .The following theorem was formulated in Section 6.3 of [KoSo1].

Theorem 2. The map Φ : H(C) → RΓ given by the formula

Φ(ν) = (ν, w)eγ , ν ∈ H(C)γis a homomorphism of Γ-graded Q-algebras. Here w is the motivic weight and (•, •)is the pairing between motivic measures and functions.

In other words, the homomorphism H(C) → RΓ can be written as

[π : Y → Ob(C)] →

→∫Y

(1−MF (π(y))) (1−MF0(Qπ(y)))L− 1

2 rkQπ(y) L12 (π(y),π(y))≤1 ecl(π(y)) ,

where∫Y

is understood as the direct image functor.

The natural extension of the above homomorphism to the completion of H(CV )maps the element AHall

V to the element AmotV .

The proof of the above theorem given in [KoSo1] is based on the motivic Thom-Sebastiani theorem and a certain identity of motivic integrals formulated in Section4.4 of loc. cit. We do not reproduce the identity here. We formulated it as theConjecture 4 and gave a proof of it in the l-adic case (i.e. when instead of the theoryof motivic functions and its equivariant version one uses the corresponding well-known theory of constructible l-adic functions). Therefore, technically speaking,the above theorem was proved in [KoSo1] only for the l-adic quantum torus andthe l-adic Hall algebra rather then for their motivic cousins. The l-adic versionshowever suffice if we are interested in numerical DT-invariants. Indeed, the resultsof Section 4.6 of [KoSo1] imply that one can use the l-adic realization of motivicDT-invariants in order to obtain the numerical DT-invariants (modulo the existenceof the quasi-classical limit, as we explain below). In order to simplify the notation

69


and exposition we are going to assume below that Theorem 2 is true in its fullgenerality (i.e. that the Conjecture 4 from [KoSo1] holds not only l-adically, butfor motives). Then we can define the collection Φ(AHall

V ) := AmotV parametrized by

strict sectors V ⊂ R2.

Definition 5. Let C be an ind-constructible 3-dimensional Calabi-Yau categoryendowed with a stability condition σ ∈ Stab(C, cl) and an orientation data. We callthe collection of elements (Amot

V ∈ RV ) of the completed motivic quantum tori (RV )(for all strict sectors V ⊂ R2) the motivic Donaldson-Thomas invariant of C.

4.4. Quasi-classical limit and integrality. Let us consider the followingunital Q-subalgebra of Q(q1/2):

Dq := Z[q1/2, q−1/2,((qn − 1)−1

)n�1

] .

There is a homomorphism of rings Dμ → Dq given by the twisted Serre polynomial.

Namely, it maps L1/2 → q1/2, and on Mμ it is the composition of the Serre poly-

nomial with the involution q1/2 → −q1/2. Therefore, we have a homomorphism ofalgebras RΓ → RΓ,q, where RΓ,q is the Dq-algebra generated by eγ , γ ∈ Γ, subjectto the relations

eγ eμ = q12 〈γ,μ〉eγ+μ , e0 = 1 .

Here the integer skew-symmetric form 〈•, •〉 on Γ is a part of the data defining astability condition on C (see Introduction in [KoSo1] for the details and explanationson why our axioms are slightly different from those of Bridgeland). Similarly to themotivic case, we have an algebra RV,q associated with any strict sector V .

The elements AV,q ∈ RV,q corresponding to AmotV are series in eγ , γ ∈ Γ with

coefficients which are rational functions in q1/2. They can have poles as qn = 1 forsome n � 1. Hence it is not clear how to take the quasi-classical limit as q1/2 → −1(this corresponds to the taking of Euler characteristic of the corresponding motives).

Let us assume that the skew-symmetric form on Γ is non-degenerate (otherwisewe can replace Γ by the symplectic lattice Γ ⊕ Γ∨). The element AV,q defines anautomorphism of an appropriate completion of RΓ,q. More precisely, it acts by the

conjugation x → AV,qxA−1V,q on the subring

∏γ∈C0(V )∩Γ

Dq eγ

where C0(V ) = C0(V, Z,Q) is the union of 0 with the convex hull C(V, Z,Q) of theset Z−1(V ) ∩ {Q � 0}.

The “integer” quantum torus⊕γ∈C0(V )∩Γ

Z[q±1/2]eγ ⊂ RΓ,q

has the quasi-classical limit1 which is the Poisson algebra with basis eγ , γ ∈ C0(V )∩Γ with product and Poisson bracket given by

eγeμ = (−1)〈γ,μ〉eγ+μ, {eγ , eμ} = (−1)〈γ,μ〉〈γ, μ〉eγ+μ .

The Poisson bracket is the limit of a normalized bracket:

[eγ , eμ] =(q1/2〈γ,μ〉 − q−1/2〈γ,μ〉

)eγ+μ ,

1There is another quasi-classical limit q1/2 → +1 which we do not consider here.

70


limq1/2→−1

(q − 1)−1 ·(q1/2〈γ,μ〉 − q−1/2〈γ,μ〉

)= (−1)〈γ,μ〉〈γ, μ〉 .

One can write informally

eγ = limq1/2→−1

eγq − 1

.

The following “absense of poles conjecture” was formulated in [KoSo1], Section7.1.

Conjecture 1. For any 3d Calabi-Yau category with a stability condition andany strict sector V the automorphism x → AV,qxA

−1V,q preserves the subring

∏γ∈C0(V )∩Γ

D+q eγ ,

where D+q := Z[q±1/2].

Furthermore, in Section 7.3 of [KoSo1] we formulated a stronger conjectureat the level of motivic Hall algebras. It seems that the above “absense of polesconjecture” holds for a larger class of categories than the class of 3CY categories.

Assuming the above conjecture we denote by AV the “quasi-classical limit” ofthe automorphism x → AV,qxA

−1V,q as q1/2 → −1. This is a symplectomorphism of

the symplectic torus associated with the symplectic lattice Γ (or, better, with thesymplectic lattice Γ⊕Γ∨). This torus can be thought of as the quasi-classical limitof the motivic quantum torus. For a generic stability condition this symplectomor-phism can be written as

AV =−→∏

Z(γ)∈V

T Ω(γ)γ ,

where

Tγ(eμ) = (1− eγ)〈γ,μ〉eμ

and Ω(γ) ∈ Q. The collection (AV ) satisfies the Factorization Property. In thesame Section 7.1 of [KoSo1] we formulated the following conjecture.

Conjecture 2. For a generic central charge Z all numbers Ω(γ), γ ∈ Γ \ {0}are integers.

We discussed non-trivial arguments in favor of this “integrality conjecture” inSection 7.5 of [KoSo1]. In the case of the category generated by one Schur objectE (i.e. Ext<0(E,E) = 0 and Hom(E,E) is one-dimensional) we reformulated it interms of the generating function of Euler characteristics of certain moduli spacesof representations of a quiver with potential. Although in general the conjectureis still open, there are some partial results confirming it. In particular, Reinekeproved in [Re1] that the integrality property is preserved under the wall-crossingformula (see below). In [KoSo3] we obtained a stronger than [Re1] integrality resultat the level of motivic DT-invariants, including the integrality in case of quiverswith potential.

Remark 2. The collection (Ω(γ))γ∈Γ seems to be the correct mathematicaldefinition of the counting of BPS states in String Theory.

71


The quasi-classical limit of motivic DT-invariants can be compared with the“microlocal” version of DT-invariants introduced by Kai Behrend in [Be1]. Recallthat he defined a Z-valued invariant of a critical point x of a function f on X whichis equal to

(−1)dimX(1− χ(MFx(f))) ,

where χ denotes the Euler characteristic. By Thom-Sebastiani theorem this numberdoes not change if we add to f a function with a quadratic singularity at x (stableequivalence).

Let M be a scheme with symmetric perfect obstruction theory (see [BeFa]).Then M is locally represented as a scheme of critical points of a function f on amanifold X. The above invariant gives rise to a Z-valued constructible function Bon M . The global invariant is

∫

M

B dχ :=∑n∈Z

nχ(B−1(n)) ,

where χ denotes the Euler characteristic. Behrend proved that for a proper Mthe invariant

∫M

B dχ coincides with the degree of the virtual fundamental class

[M ]virt ∈ H0(M) given by∫[M ]virt 1.

Now let us assume that M ⊂ Css consists of Schur objects. For such an objectE let cl(E) = γ ∈ Γ be a fixed primitive class. Then the contribution of M intothe γ-component of the motivic DT-invariant is equal to

∫

M

L12 (1−dimExt1(E,E))

L− 1(1−MF (E))(1−MF0(QE))L

− 12 rkQE eγ .

In the quasi-classical limit this contribution is easily seen to be equal to −Ω(γ).Hence Behrend formula implies that the contribution of M to the value Ω(γ) isequal to

∫[M ]virt 1.

Remark 3. An important property of any generalization of DT-invariants istheir deformation invariance. In Section 7.2 of [KoSo1] we formulated a conjec-ture which claims the invariance of the collection (Ω(γ))γ∈Γ with respect to the“polarization preserving” deformations of C, in the case when C is homologicallysmooth (the latter is the categorical analog of proper smooth schemes introduced in[KoSo2]). In the geometric framework (i.e. for the category of coherent sheaves)there is a theory of generalized DT-invariants developed by Joyce and Song (see[JoS1,2]). In this more restrictive setting they can prove deformation invarianceproperty of their invariants.

5. Stability data on graded Lie algebras and wall-crossing formulas

In Section 2 of [KoSo1] we developed a general approach to “numerical” DTinvariants, which is independent on the Calabi-Yau property and even of categories.It is based on the notion of stability data on graded Lie algebras.

5.1. Stability data. Let us fix a free abelian group Γ of finite rank, and agraded Lie algebra g = ⊕γ∈Γgγ over Q.2

2In examples g is a R-linear Lie algebra, where R is a commutative unital Q-algebra.

72


Definition 6. A stability data on g is a pair σ = (Z, a) such that:1) Z : Γ → R2 � C is a homomorphism of abelian groups called the central

charge;2) a = (a(γ))γ∈Γ\{0} is a collection of elements a(γ) ∈ gγ ,satisfying the following

Support Property:

Pick a norm ‖ • ‖ on the vector space ΓR = Γ⊗Z R. Then there exists C > 0such that for any γ ∈ Supp a (i.e. a(γ) �= 0) one has

‖ γ ‖≤ C|Z(γ)| .

Obviously the Support Property does not depend on the choice of the norm.We will denote the set of all stability data on g by Stab(g). Later we will equipthis set with a Hausdorff topology.

The Support Property is equivalent to the following condition (which we willalso call the Support Property):

There exists a quadratic form Q on ΓR such that1) Q|Ker Z < 0;2) Supp a ⊂ {γ ∈ Γ \ {0}| Q(γ) � 0},where we use the same notation Z for the natural extension of Z to ΓR.Indeed, we may assume that the norm ‖ • ‖ is the Euclidean norm in a chosen

basis and take Q(γ) = − ‖ γ ‖2 +C1|Z(γ)|2 for sufficiently large positive constantC1. Generically Q has signature (2, n− 2), where n = rkΓ.

For a given quadratic form Q on ΓR we denote by StabQ(g) ⊂ Stab(g) the setof stability data satisfying the above conditions 1) and 2). Obviously Stab(g) =∪QStabQ(g), where the union is taken over all quadratic forms Q.

In Section 2.2 of [KoSo1] we reformulated the stability data in the followingequivalent form.

We denote by StabQ(g) the set of pairs (Z,A) such that:a) Z : Γ → R2 is an additive map such that Q|KerZ < 0;b) A = (AV )V ∈S is a collections of elements AV ∈ GV,Z,Q, where GV,Z,Q is a

pronilpotent group with the pronilpotent graded Lie algebra

gV,Z,Q =∏

γ∈Γ∩C(V,Z,Q)

gγ ,

where C(V, Z,Q) is the convex cone generated by the set

S(V, Z,Q) = {x ∈ ΓR \ {0}|Z(x) ∈ V,Q(x) � 0} .

For a triangle Δ which is cut out of the sector V by a straight line, any γ ∈Z−1(Δ) can be represented as a sum of elements of Γ∩C(V, Z,Q) in finitely manyways. Furthermore, the triangle Δ defines the Lie ideal JΔ ⊂ gV,Z,Q consisting ofelements y = (yγ) ∈ gV,Z,Q such that for every component yγ the corresponding γdoes not belong to the convex hull of Z−1(Δ). Then the quotient gΔ := gV,Z,Q/JΔis a nilpotent Lie algebra, and gV,Z,Q = lim←−Δ⊂V

gΔ. Let GΔ = exp(gΔ) be the

pronilpotent group with the pronilpotent Lie algebra gΔ. Then GV,Z,Q = lim←−ΔGΔ

is a pronilpotent group. If V = V1 � V2 (in the clockwise order) then there arenatural embeddings GVi,Z,Q → GV,Z,Q, i = 1, 2. We impose the following axiom onthe set of pairs (Z,A):

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Factorization Property:The element AV is given by the product AV = AV1

AV2where the equality is

understood in GV,Z,Q.If we assume the “quasi-classical limit” conjecture discussed in Section 4.4 then

the algebra of functions on the symplectic torus (Γ⊕ Γ∨)⊗R/Γ⊕ Γ∨ gives rise toa Γ⊕Γ∨-graded Lie algebra. The collection of symplectomorphisms (AV ) obtainedas the limit of the automorphisms (Ad(AV,q)) as q1/2 → −1 gives rise to stabilitydata on this graded Lie algebra.

5.2. Wall-crossing formulas. In Section 2.3 of [KoSo1] we defined a Haus-dorff topology on the space Stab(g). Let us recall this definition here. Let Xbe a topological space, x0 ∈ X be a point, and (Zx, ax) ∈ Stab(g) be a familyparametrized by X.

Definition 7. We say that the family is continuous at x0 if the followingconditions are satisfied:

a) The map X → Hom(Γ,C), x → Zx is continuous at x = x0.b) If a quadratic form Q0 such that (Zx0

, ax0) ∈ StabQ0

(g) then there exists anopen neighborhood U0 of x0 such that (Zx, ax) ∈ StabQ0

(g) for all x ∈ U0.c) For any closed strict sector V such that Z(Supp ax0

) ∩ ∂V = ∅ the map

x → log AV,x,Qx∈ gV,Zx,Qx

⊂∏γ∈Γ

gγ

is continuous at x = x0. Here we endow the vector space∏

γ∈Γ gγ with the product

topology of discrete sets, and AV,x,Qxis the group element associated with (Zx, ax),

the sector V and a quadratic form Qx such that (Zx, ax) ∈ StabQx(g).

If g arises from a 3CY category then this definition has a nice categoricalinterpretation given in Section 3.4 of loc. cit. Having a topology defined in sucha way one immediately arrives at the wall-crossing formulas for the stability datasuch as follows.

Let us fix an element Z0 ∈ Hom(Γ,C) and a quadratic form Q0 compatible withZ0 (i.e. negative on its kernel). We denote by UQ0,Z0

the connected componentcontaining Z0 in the domain {Z ∈ Hom(Γ,C)| (Q0)|KerZ < 0}. Let γ1, γ2 ∈ Γ\{0}be two Q-linearly independent elements such that Q0(γi) � 0, Q0(γ1 + γ2) � 0, i =1, 2. We introduce the set

WQ0γ1,γ2

= {Z ∈ UQ0,Z0| R>0Z(γ1) = R>0Z(γ2)} .

In this way we obtain a countable collection of real hypersurfaces WQ0γ1,γ2

⊂ UQ0,Z0

called the walls corresponding to Q0, γ1, γ2. We denote their union by W1 := WQ0

1

and call it the wall of first kind (physicists call it the wall of marginal stability).Let us consider a continuous path Zt, 0 ≤ t ≤ 1 in UQ0,Z0

which intersectseach of these walls for finitely many values of t ∈ [0, 1]. Suppose that we have acontinuous lifting path (Zt, at) of this path in Stab(g) such that Q0 is compatiblewith each at for all 0 ≤ t ≤ 1. Then for any γ ∈ Γ \ {0} such that Q0(γ) � 0 theelement at(γ) does not change as long as t satisfies the condition

Zt(γ) /∈ ∪γ1,γ2∈Γ\{0}, γ1+γ2=γWQ0γ1,γ2

.

If this condition is not satisfied we say that t is a discontinuity point for γ. For agiven γ there are finitely many discontinuity points.

74


Notice that for each t ∈ [0, 1] there exist limits

a±t (γ) = limε→0, ε>0

at±ε(γ)

(for t = 0 or t = 1 only one of the two limits is well-defined). Then the continuityof the lifted path (Zt, at) is equivalent to the following wall-crossing formula whichholds for any t ∈ [0, 1] and arbitrary γ ∈ Γ \ {0}:

−→∏μ∈Γprim, Zt(μ)∈lγ,t

exp

⎛⎝∑

n�1

a−t (nμ)

⎞⎠ =

= exp

⎛⎝ ∑

μ∈Γprim, Zt(μ)∈lγ,t, n�1

at(nμ)

⎞⎠ =

−→∏μ∈Γprim, Zt(μ)∈lγ,t

exp

⎛⎝∑

n�1

a+t (nμ)

⎞⎠ ,

where lγ,t = R>0Zt(γ), and Γprim ⊂ Γ is the set of primitive vectors. The firstand the last products are taken in the clockwise order of Arg(Zt−ε) and Arg(Zt+ε)respectively, where ε > 0 is sufficiently small. Moreover, for each γ we have a−t (γ) =a+t (γ) = at(γ) unless there exist non-zero γ1, γ2 such that γ = γ1 + γ2 and Zt ∈WQ0

γ1,γ2.

Remark 4. Informally speaking, the wall-crossing formula says that for a verysmall sector V containing the ray lγ,t the corresponding element AV , considered asa function of time, is locally constant in a neighborhood of t.

For each γ ∈ Γ \ {0} the wall-crossing formula allows us to calculate a1(γ) isterms of a0(γ

′) for a finite collection of elements γ′ ∈ Γ \ {0}. Morally it is aninductive procedure on the ordered set of discontinuity points ti ∈ [0, 1].

In the case of a 3CY category the wall-crossing formula can be written in avery explicit form. For that it is not necessary to enlarge the “charge lattice” Γ.More precisely, let Γ be a free abelian group of finite rank n, endowed with a skew-symmetric integer-valued bilinear form 〈•, •〉 : Γ×Γ → Z. Then the correspondingLie algebra of functions on a Poisson torus (which is the quasi-classical limit of thequantum torus associated with the 3CY category) is given explicitly in terms ofgenerators and relations as gΓ = gΓ,〈•,•〉 = ⊕γ∈ΓQ · eγ with the Lie bracket

[eγ1, eγ2

] = (−1)〈γ1,γ2〉〈γ1, γ2〉eγ1+γ2.

It can be made into a Poisson algebra with the commutative prouct given byeγ1

eγ2= (−1)〈γ1,γ2〉 eγ1+γ2

.For a stability data (Z, a) we can write uniquely (by the Mobius inversion

formula)

a(γ) = −∑

n�1, 1nγ∈Γ\{0}

Ω(γ/n)

n2eγ ,

where Ω : Γ \ {0} → Q is a function. Then we have

exp

⎛⎝∑

n�1

a(nγ)

⎞⎠ = exp

⎛⎝−

∑n�1

Ω(nγ)∑k�1

eknγk2

⎞⎠ := exp

⎛⎝−

∑n�1

Ω(nγ) Li2(enγ)

⎞⎠ ,

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where Li2(t) =∑

k�1tk

k2 is the dilogarithm function. Then

AV :=−→∏

γ∈C(V,Z)∩Γ

exp

(−Ω(γ)

∞∑n=1

enγn2

).

If the stability data belongs to the wall of first kind then there exists a 2-dimensional lattice Γ0 ⊂ Γ such that Zt0(Γ0) belongs to a real line Reiα for someα ∈ [0, π].

The wall-crossing formula describes the change of the values Ω(γ) for γ ∈ Γ0

and depends only on the restriction Ω|Γ0of Ω to the lattice Γ0. Values Ω(γ) for

γ /∈ Γ0 do not change at t = t0. Denote by k ∈ Z the value of the form 〈•, •〉 on afixed basis γ1, γ2 of Γ0 � Z2 such that C(V )∩Γ0 ⊂ Z�0 ·γ1⊕Z�0 ·γ2, where C(V )is the convex cone defined previosuly. We assume that k �= 0, otherwise there willbe no jump in values of Ω on Γ0. The group elements which we are interested in canbe identified with products of the following automorphisms of Q[[x, y]] preservingthe symplectic form k−1(xy)−1dx ∧ dy:

T(k)a,b : (x, y) →

→(x · (1− (−1)kabxayb)−kb, y · (1− (−1)kabxayb)ka

), a, b � 0, a+ b � 1 .

For γ = aγ1 + bγ2 we have

T(k)a,b = exp

⎛⎝−

∑n�1

enγn2

⎞⎠

in the above notation. Any exact symplectomorphism φ of Q[[x, y]] can be decom-posed uniquely into a clockwise and an anti-clockwise product:

φ =−→∏a,b

(T

(k)a,b

)ca,b

=←−∏a,b

(T

(k)a,b

)da,b

with certain exponents ca,b, da,b ∈ Q. These exponents should be interpreted as thelimiting values of the functions Ω±

t0 = limt→t0±0 Ωt restricted to Γ0. The passage

from the clockwise order (when the slope a/b ∈ [0,+∞] ∩ P1(Q) decreases) to the

anti-clockwise order (when the slope increases) gives the change of Ω|Γ0as we cross

the wall. It will be convenient to denote T(1)a,b simply by Ta,b. The pronilpotent

group generated by the transformations T(k)a,b coincides with the one generated by

the transformations Ta,|k|b.For instance the decompositions for k = 1, 2 look such as follows:

T1,0 · T0,1 = T0,1 · T1,1 · T1,0 ,

T(2)1,0 · T (2)

0,1 = T(2)0,1 · T (2)

1,2 · T (2)2,3 · · · · · (T (2)

1,1 )−2 · · · · · T (2)

3,2 · T (2)2,1 · T (2)

1,0 ,

or equivalently

T1,0 · T0,2 = T0,2 · T1,4 · T2,6 · · · · · T−21,2 · · · · · T3,4 · T2,2 · T1,0 .

The formula for k = 2 is related to the one for the BPS spectrum in the pureN = 2, d = 4 Seiberg-Witten theory.

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5.3. Walls of second kind. Let us fix as above the quadratic form Q0 and aconnected component U of the set {Z ∈ Hom(Γ,C)| (Q0)|KerZ < 0}. For a given

primitive γ ∈ Γ \ {0} we introduce the set WQ0γ = {Z ∈ U |Z(γ) ∈ R>0}. It is a

real hypersurface in U . We call it the wall of second kind associated with γ. Wecall the union ∪γWQ0

γ the wall of second kind and denote it by W2.

Definition 8. We say that a path σ = (Zt)0≤t≤1 ⊂ U is short if the convex

cone Cσ which is the convex hull of(∪0≤t≤1Z

−1t (R>0)

)∩ {Q0 � 0} is strict.

With a short path we associate a pronilpotent group GCσwith Lie algebra

gCσ=

∏γ∈Cσ∩Γ gγ .

It follows from the definition of the topology that there is a lifting map φ :U → Stab(g) (more precisely, one uses the fact that the projection of the space ofstability conditions to the space of central charges is a local homeomorphism).

Then one can easily prove the following result.

Proposition 2. For a generic short path σ = (Zt)0≤t≤1 there exists a nomore than countable set ti ∈ [0, 1] and corresponding primitive γi ∈ Γ \ {0} suchthat Zti ∈ WQ0

γi. For each i we have: rkZ−1

ti (R) ∩ Γ = 1.

For every such ti we introduce

Ati = exp

⎛⎝εi

∑n�1

ati(nγi)

⎞⎠ ∈ GCσ

,

where εi = ±1 depending on the direction in which the path Zt(γi) crosses R>0

for t sufficiently close to ti.Then the wall-crossing formulas imply the following result (Theorem 4 from

Section 2.4 of [KoSo1]).

Theorem 3. For any short loop the monodromy∏−→

tiAti is equal to the iden-

tity (here the product is taken in the increasing order of the elements ti).

This “triviality of the monodromy” allows us to introduce (at least for a shortpath) the well-defined notion of parallel transport. In the categorical frameworkthe parallel transport appears when one calculates the change of the “numerical”DT-invariants under the change of t-structure (e.g. under tilting).

6. Examples

6.1. Quivers with potential and cluster transformations. Let C be a3-dimensional ind-constructible Calabi-Yau category over a field k of characteristiczero. Assume that it is endowed with a finite collection of spherical generators E ={Ei}i∈I of C defined over k. Then Ext•C(k)(Ei, Ei) is isomorphic toH•(S3,k), i ∈ I.

The matrix of the Euler form (taken with the minus sign)

aij := −χ(Ext•C(k)(Ei, Ej)

)

is integer and skew-symmetric. In fact, the ind-constructible category C can becanonically derived from the k-linear Calabi-Yau A∞-category C(k), or even fromits full subcategory consisting of the collection E . In what follows we will omit thesubscript C(k) in the notation for Ext•-spaces.

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Definition 9. The collection E is called a cluster if for any i �= j the gradedspace ⊕m∈Z Extm(Ei, Ej) is either zero, or it is concentrated in one of the twodegrees m = 1 or m = 2 only.

We will assume that our collection is a cluster. In that case K0(C(k)) � ZI

with the basis formed by the isomorphism classes [Ei], i ∈ I. With a cluster collec-tion one associates a quiver Q with potential W in the natural way: the verticesof Q correspond to the cluster generators, the arrows between vertices i and jcorrespond to a basis in Ext1(Ei, Ej) and the potential W is the restriction of the

potentail of the category to the ⊕i,j∈I Ext1(Ei, Ej). It is proved in [KoSo1], Section

8.1, Theorem 9 that the pairs (Q,W ) up to a naturally defined gauge equivalencecorrespond to 3CY categories (up to equivalence) generated by a cluster collectionfor which

• Ext0(Ei, Ei) = k idEi,

• Ext0(Ei, Ej) = 0 for any i �= j,

• Ext<0(Ei, Ej) = 0, for any i, j.

Mutation (a.k.a. spherical reflection) in C corresponds to the notion of muta-tion of (Q,W ) in quiver theory. Cluster generators single out an open domain inStab(C), where all central charges Z(Ei) belong to the upper half-plane. Categor-ical mutations change the t-structure, so that some Z(Ei) can move to the lowerhalf-plane. It can be geometrically interpreted as a rotation of the upper-half planeand hence corresponds to the crossing of the wall of second kind. The motivic DT-invariant gives rise to an automorphism of the corresponding quantum torus, as wediscussed before. It can be written in terms of the quiver (Q,W ) as well as in termsof the mutated quiver (Q′,W ′). Two expressions of the same motivic DT-invariantare related by a quantum cluster transformation (see [KoSo1], Section 8.4 Corollary4 for a precise statement). It would be interesting to compare this picture with theone developed in [GaMoNe2].

6.2. Stability data from complex integrable systems. In Section 2.7 of[KoSo1] we explained how one can associate stability data on a graded Lie algebrato a complex integrable system. Recall that a complex integrable system is a holo-morphic map π : X → B where (X,ω2,0

X ) is a holomorphic symplectic manifold,dimX = 2dimB, and the generic fiber of π is a Lagrangian submanifold, which isa polarized abelian variety. We assume (in order to simplify the exposition) thatthe polarization is principal. The fibration π is non-singular outside of a closedsubvariety Bsing ⊂ B of complex codimension at least one. It follows that on theopen subset Bsm := B \Bsing we have a local system Γ of symplectic lattices withthe fiber over b ∈ Bsm equal to Γb := H1(Xb,Z), Xb = π−1(b) (the symplecticstructure on Γb is given by the polarization).

Furthermore, the set Bsm is locally (near each point b ∈ Bsm) embedded asa holomorphic Lagrangian subvariety into an affine symplectic space parallel toH1(Xb,C). Namely, let us choose a symplectic basis γi ∈ Γb, 1 ≤ i ≤ 2n. Then

we have a collection of holomorphic closed 1-forms αi =∫γiω2,0X , 1 ≤ i ≤ 2n in a

neighborhood of b. There exists (well-defined locally up to an additive constant)holomorphic functions zi, 1 ≤ i ≤ 2n such that αi = dzi, 1 ≤ i ≤ 2n. Theydefine an embedding of a neighborhood of b into C2n. The collection of 1-formsαi gives rise to an element δ ∈ H1(Bsm,Γ∨ ⊗ C). We assume that δ = 0. This

78


assumption is equivalent to an existence of a section Z ∈ Γ(Bsm,Γ∨ ⊗OBsm) suchthat αi = dZ(γi), 1 ≤ i ≤ 2n.

Definition 10. We call Z the central charge of the integrable system.

Hence, for every point b ∈ Bsm we have a symplectic lattice Γb endowed withan additive map Zb : Γb → C. In the loc. cit. we defined a continuous family ofstability data on graded Lie algebras gΓb

with central charges Zb in the followingway.

First, observe that the dense open set Bsm ⊂ B carries a Kahler form

ω1,1B = Im

⎛⎝ ∑

1≤i≤n

αi ∧ αn+i

⎞⎠ .

We denote by gB the corresponding Kahler metric.For any t ∈ C∗ we define an integral affine structure on the C∞-manifold

Bsm given by a collection of closed 1-forms Re(tαi), 1 ≤ i ≤ 2n. For any simply-connected open subset U ⊂ Bsm and a covariantly constant section γ ∈ Γ(U,Γ) wehave a closed 1-form

αγ,t = Re

(t

∫γ

ω2,0X

)= dRe(tZ(γ)) ,

and the corresponding gradient vector field vγ,t = g−1B (αγ,t). Notice that this

vector field is a constant field with integral direction in the integral affine structureassociated with the closed 1-forms Im(tαi), 1 ≤ i ≤ 2n. Second, using the approachof [KoSo4] we can construct an infinite oriented tree lying in B such that its externalvertices belong to Bsing, and its edges are positively oriented trajectories of thevector fields vγ,t. All internal vertices have valency at least 3, and every such vertexshould be thought of as a splitting point: a trajectory of the vector field vγ,t is splitat a vertex into several trajectories of some vector fields vγ1,t, . . . , vγk,t such thatγ = γ1 + · · ·+ γk.

For a fixed t ∈ C∗ the union Wt of all trees as above is in fact a countable unionof real hypersurfaces in Bsm. They are analogs of the walls of second kind. The setWt depends on Arg t only. The union ∪θ∈[0,2π)Wteiθ swaps the whole space Bsm.

Let us denote by W (1) the union over all t ∈ C∗/R>0 of the sets of internal verticesof all trees in Wt (splitting points of the gradient trajectories). This is an analog ofthe wall of first kind. Once again using the approach of [KoSo4] we assign rationalmultiplicities to edges of the tree. This leads to the following picture. Considerthe total space tot(Γ) of the local system Γ. It follows from above assumptionsand considerations that we have a locally constant function Ω : tot(Γ) → Q whichjumps at the subset consisting of the lifts of the wall W (1) to tot(Γ). Then for afixed b ∈ Bsm the pair (Z,Ω) defines stability data on the graded Lie algebra gΓb

of the group of formal symplectomorphisms of the symplectic torus associated withΓb. In this way we obtain a local embedding Bsm ↪→ Stab(gΓb

).Examples of this construction include integrable systems of Seiberg-Witten

theory, Hitchin systems, etc. The case of pure SU(2) Seiberg-Witten theory wasillustrated in Section 2.7 of [KoSo1]. The corresponding wall-crossing formula co-

incides with the one for T(2)a,b considered above. More examples can be found in

[GaMoNe1,2]. We also remark that this section is related to Sections 7.2, 7.6 be-low.

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6.3. Cecotti-Vafa work and WCF for gl(n). This is an example of thewall-crossing formula considered in [KoSo1], Section 2.9.

Let g = gl(n,Q) be the Lie algebra of the general linear group. We consider itas a Γ-graded Lie algebra g = ⊕γ∈Γgγ , where

Γ = {(k1, . . . , kn)| ki ∈ Z,∑

1≤i≤n

ki = 0}

is the root lattice. We endow g with the Cartan involution η. The algebra g hasthe standard basis Eij ∈ gγij

consisting of matrices with the single non-zero entryat the place (i, j) equal to 1. Then η(Eij) = −Eji. In [KoSo1], Section 2.1 weintroduced also the notion of symmetric stability data on a graded Lie algebra.The definition includes a choice of involution η on Γ. In this section we are goingto consider symmetric stability data.

We notice that

Hom(Γ,C) � Cn/C · (1, . . . , 1) .We define a subspace Hom◦(Γ,C) ⊂ Hom(Γ,C) consisting (up to a shift by themultiples of the vector (1, . . . , 1)) of vectors (z1, . . . , zn) such that zi �= zj if i �= j.Similarly we define a subspace Hom◦◦(Γ,C) ⊂ Hom(Γ,C) consisting (up to thesame shift) of such (z1, . . . , zn) that there is no zi, zj , zk belonging to the samereal line as long as i �= j �= k. Obviously there is an inclusion Hom◦◦(Γ,C) ⊂Hom◦(Γ,C).

For Z ∈ Hom(Γ,C) we have Z(γij) = zi − zj . If Z ∈ Hom◦◦(Γ,C) thensymmetric stability data with such Z is the same as a skew-symmetric matrix(aij) with rational entries determined from the equality a(γij) = aijEij . Everycontinuous path in Hom◦(Γ,C) admits a unique lifting to Stab(g) as long as wefix the lifting of the initial point. The matrix (aij) changes when we cross wallsin Hom◦(Γ,C) \ Hom◦◦(Γ,C). A typical wall-crossing corresponds to the casewhen in the above notation the point zj crosses a straight segment joining zi andzk, i �= j �= k. In this case the only change in the matrix (aij) is of the form:

aik → aik + aijajk .

Exactly these wall-crossing formulas appeared in [CeVa] (the numbers aij are inte-gers in the loc. cit.)

6.4. About explicit formulas. Assume that a 3-dimensional Calabi-Yaucategory C is generated by one spherical object E defined over k. ThereforeR := Ext•(E,E) � H•(S3,k). In this case we take Γ = K0(C(k)) � Z, andthe skew-symmetric form on Γ is trivial. We explained the choice of orinetationdata in [KoSo1], Section 6.4. For any z ∈ C, Im z > 0 we have a stability conditionσz such that E ∈ Css, Z(E) := Z(clk(E)) = z, Arg(E) = Arg(z) ∈ (0, π). For astrict sector V such that Arg(V ) ⊂ (0, π) we have the category CV which is eithertrivial (if z /∈ V ) or consists of objects 0, E,E ⊕ E, . . . (if z ∈ V ). Then Amot

V = 1in the first case and

AmotV =

∑n�0

Ln2/2

[GL(n)]enγ1

,

in the second case. Here γ1 is the generator of Γ (i.e. the class of E in theK-theory).In this case Ext1(nE, nE) = 0, where we set nE = E⊕n, n � 1. Therefore

WnE = 0 which implies that for the motivic Milnor fiber we have MF (WnE) = 0.

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The numerator in the above formula is

Ln2/2 = L

12 dimExt0(nE,nE) = L

12

∑i≤1(−1)i dimExti(nE,nE) ,

since Ext1(nE, nE) = 0,Ext<0(nE, nE) = 0.Let us consider the “quantum dilogarithm” series

E(q1/2, x) =∑n�0

qn2/2

(qn − 1) . . . (qn − qn−1)xn ∈ Q(q1/2)[[x]] .

Since [GL(n)] = (Ln − 1) . . . (Ln − Ln−1), we conclude that

AmotV = E(L1/2, eγ1

) .

In Section 6.4 of [KoSo1] we presented wall-crossing formulas for both motivicand numerical DT-invariants in several cases (e.g. for D0-D6 BPS bound states).There are many more computations of this sort in the literature. Most of them arerelated to toric local 3CY varieties and the count often reduces to the count of cyclicmodules similarly to [KoSo1], Sections 7.3,7.4. We mention here only few paperswhich contain explicit formulas: [DeMo], [GuDi], [Jaf], [JafMo], [NagNak],[Nag1],[Nag2], [OoYa], [Sz].

7. Discussion, speculations, open questions

There are several foundational questions in the theory of motivic DT-invariantswhich have to be settled, e.g. deformation invariance, existence of orientation data(or its invariance under mutations in the case of quivers with potential), existenceof the quasi-classical limit, proof of the integral identity in the general motiviccase. All those questions were addressed in [KoSo1]. We would like to recall belowseveral other open problems discussed in [KoSo1] which, in our opinion, deservefurther study.

7.1. Intermediate Jacobian, complex integrable systems for 3CY andGW=DT. The idea of associating a complex integrable system to a homologicallysmooth 3CY category was discussed in [KoSo1], Section 7.2. It is partially moti-vated by the corresponding pure geometric story (see [DonMar]). In addition tothat we speculated on how Donaldson-Thomas (and Gromov-Witten) invariantscould emerge in this case.

More precisely, we suggested that for an arbitrary triangulated compact homo-logically smooth A∞-category C (see [KoSo2]) one has a non-commutative versionof the Deligne cohomology HD(C) which fits into a short exact sequence definedvia the Hodge filtration on periodic cyclic homology HP•(C) (we also use the hy-

pothetical topological K-group Ktop• ):

0 → HPodd(C)/(F 1/2odd +Ktop

odd(C)) → HD(C) → F 0even ∩Ktop

even(C) → 0 .

Morally, HD(C) should be thought of as zero cohomology group of the homotopycolimit of the following diagram of cohomology theories:

HC−• (C)⏐⏐�

Ktop• (C) −−−−→ HP•(C)

where HC−• (C) is the negative cyclic homology.

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Any object of C should have its characteristic class in HD(C). More precisely,there should be a homomorphism of groups chD : K0(C) → HD(C) (in the case ofCalabi-Yau manifolds it is related to holomorphic Chern-Simons functional). Thereason for this is that every object E ∈ Ob(C) should have natural characteristic

classes in Ktop0 (C) and in HC−

0 (C) whose images in HP0(C) coincide with eachother.

Let now C be a 3CY category. Then we have the moduli space M of itsdeformations (including the Calabi-Yau structure). We conjectured in [KoSo1]that there is a fibration Mtot → M with the fiber HD(C) over the point [C] ∈ Msuch that Mtot is a holomorphic symplectic manifold. Moreover, any fiber of thisfibration (i.e. the group HD(C) for given [C]) is a countable union of complexLagrangian tori. By analogy with the commutative case we suggested that thelocus L ⊂ Mtot consisting of values of chD is a countable union of Lagrangiansubvarieties. Every such subvariety can be either a finite ramified covering of M ora fibration over a proper subvariety of M with the fibers which are abelian varieties.

For generic [C] ∈ M one can use the triple (K0(C), HD(C), chD) as a triple(K0(C),Γ, cl). Analogs of our motivic Donaldson-Thomas invariants Amot

V ∈ RV

will be formal countable sums of points in HD(C) with “weights” which are elements

of the motivic ring Dμ. The pushforward map from HD(C) to Γ = F 0even ∩Ktop

0 (C)gives the numerical DT-invariants. The continuity of motivic DT-invariants meansthat after taking the quasi-classical limit the weights become integer-valued func-tions on the set of those irreducible components of L which are finite ramifiedcoverings on M.

These considerations lead to the following question raised in [KoSo1], Section7.2

Question 1. Is there a natural extension of the numerical DT-invariants tothose components of L which project to a proper subvariety of M?

The theory of Gromov-Witten invariants can be (again hypothetically) ex-pressed in a similar way. This leads to a natural question about the generalizationof the famous conjecture “GW=DT”. The following remark from the loc. cit. canbe considered as a proposal of such generalization in the geometric case.

Remark 5. Suppose X is a 3d complex compact Calabi-Yau manifold withH1(X,Z) = 0. Then we have an exact sequence

0 → H3DR(X)/(F 2H3

DR(X) +H3(X,Z)) → H4D(X) → H4(X,Z) → 0 ,

where H4D(X) = H4(X,Z → OX → Ω1

X) is the Deligne cohomology. Then anycurve C ⊂ X defines the class [C] ∈ H4

D(X). For a generic complex structure onX the class is constant in any smooth connected family of curves. Moreover, a stablemap to X defines a class in H4

D(X). Then we have exactly the same picture withholomorphic symplectic fibration Mtot → M and Lagrangian fibers, as we discussedabove. Similarly to the case of DT-invariants the GW-invariants appear as infinitelinear combinations of points in H4

D(X), but this time with rational coefficients.We expect that the well-known relationship “GW=DT” should be a statement aboutthe equality of the above-discussed counting functions (assuming positive answer tothe Question 1).

7.2. Non-archimedean integrable systems. In [KoSo1], Section 1.5 wesuggested that the moduli space of stability conditions (or, rather a Lagrangian

82


cone in it with the induced affine structure) should be thought of as a base of non-archimedean analytic integrable system similar to the one considered in [KoSo4].Indeed, the collection of formal symplectomorphisms AV encoding the numericalDT-invariants give rise to a rigid analytic space X an over any non-archimedean field,similarly to [KoSo4]. This space carries an analytic symplectic form and describes“the behavior at infinity” of a (possibly non-algebraic) formal smooth symplecticscheme over Z. Motivated by the String Theory we conjectured that there exists anactual complex symplectic manifold M (vector or hyper multiplet moduli space)admitting a (partial) compactification M and such that its C((t))-points are givenby

X an(C((t))) = M(C[[t]]) \ (M(C[[t]]) ∪ (M\M)(C[[t]])) ,

i.e. it is the space of formal paths hitting the compactifying divisor but not belong-ing to it). In the case when C is the Fukaya category of a complex 3d Calabi-Yaumanifold X the space M looks “at infinity” as a deformation of a complex sym-plectic manifold Mcl where dimM = dimMcl = dimH3(X,C). The latter is thetotal space of the bundle Mcl → MX , where MX is the moduli space of complexstructures on X. The fiber of the bundle is isomorphic to the space

(H3,0(X) \ {0})× (H3(X,C)/H3,0(X)⊕H2,1(X)⊕H3(X,Z))

parametrizing pairs (holomorphic volume element, point of the intermediate Jaco-bian). As we explained above, we expect that there is a complex integrable systemassociated with an arbitrary homologically smooth 3d Calabi-Yau category. Thefiber is the torus associated with the “Deligne cohomology” of the category. Insome cases (see [GaMoNe1,2]) the total space carries more structures. We discusssome of them below in Section 7.6. Furthermore, the total space (constructed in[KoSo4] as a non-archimedean analytic space) can be quantized (see [So1]). Itwould be interesting to reveal the physical meaning of this quantization at the levelof hyperkahler manifolds considered in [GaMoNe1,2].

7.3. Motivic DT invariants and Cohomological Hall algebra. An al-ternative approach to motivic DT-invariants was suggested in [KoSo3]. Technicallyspeaking, it is presented in [KoSo3] in the case of quiver with potential only. Butin fact the approach of [KoSo3] is more general, with the main formalism validfor so-called smooth representation towers introduced in the loc. cit. The latterstructure is probably hidden in all 3CY categories which appear “in nature”. Sinceit is not our purpose to explain here the results of [KoSo3], we will only illustratethem in the case of a quiver without potential.

Let us fix a finite quiver Q, with the set I of vertices, and aij ∈ Z≥0 arrowsfrom i to j for i, j ∈ I. For any dimension vector

γ = (γi)i∈I ∈ ZI≥0

we have the space of representations of Q in complex coordinate vector spaces ofdimensions (γi)i∈I :

Mγ �∏i,j∈I

Caijγiγj

.

This space is naturally endowed with the conjugate action of the complex al-gebraic group

Gγ :=∏i∈I

GL(γi,C).

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We use the standard model Gr(d,C∞) := lim−→Gr(d,CN ), N → +∞ of the

classifying space of GL(d,C) for d ≥ 0, and define

BGγ :=∏i∈I

BGL(γi,C) =∏i∈I

Gr(γi,C∞) .

Let us consider the universal family over BGγ

Munivγ := (EGγ ×Mγ) /Gγ ,

where EGγ → BGγ is the universal Gγ-bundle. We introduce a ZI≥0-graded abelian

group

H := ⊕γHγ ,

where each component is defined as an equivariant cohomology

Hγ := H•Gγ

(Mγ) := H•(Munivγ ) = ⊕i≥0H

i(Munivγ ) .

Using naturally defined correspondences for the spacesMγ we defined in [KoSo3]an associative product on H, making it into cohomological Hall algebra (it shouldbe called “generalized algebra of BPS states” in physics, since it is a mathematicalincarnation of the BPS algebra discussed in [HaMo]). Toric localization gives an ex-plicit formula for the product, which makes H into a special case of Feigin-Odesskyshuffle algebra.

Let Q = Qd be now a quiver with just one vertex and d ≥ 0 loops. Then theproduct formula specializes to the following one:

(f1 · f2)(x1, . . . , xn+m) :=

∑i1<···<inj1<···<jm

{i1,...,in,j1,...,jm}=={1,...,n+m}

f1(xi1 , . . . , xin) f2(xj1 , . . . , xjm)

(n∏

k=1

m∏l=1

(xjl − xik)

)d−1

for symmetric polynomials, f1 in n variables, and f2 in m variables. The productf1 · f2 is a symmetric polynomial in n+m variables.

One can introduce a double grading on this algebra, by declaring for a ho-mogeneous symmetric polynomial of degree K in n variables to have bigrading(n, 2K + (1 − d)n2). Equivalently, one can shift the cohomological grading inH•(BGL(n,C)) by [(d− 1)n2].

For general d, the Hilbert-Poincare series Pd = Pd(z, q1/2) of the bigraded al-

gebra H is the generalized q-exponential function (a.k.a. as quantum dilogarithm):

∑n≥0,m∈Z

dim(Hn,m) znqm/2 =∑n≥0

q(1−d)n2/2

(1− q) . . . (1− qn)zn ∈ Z((q1/2))[[z]] .

Thus we see that

Al(x, q) = Pd(x, q−1) .

Here in the LHS we consider the input of the ray l in the upper-half plane tothe motivic DT-invariants of the 3CY category with the heart consisting of thefinite-dimensional representations of Q. In the case of non-zero potential the heartconsists of its critical points, and one should use the cohomology of the perversesheaf of vanishing cycles associated with the potential (see [KoSo3] for details).

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The cohomological Hall algebra does not depend on a choice of a stability condition(more precisely, it depends on a t-structure, but not on a central charge). As weshow in [KoSo3] such a choice gives rise to a PBW-type basis in H. Moreover italso gives rise to a factorization of AV into the product of factors of the type Ass

γj,

where γj is a non-zero primitive element of ZI , and each factor Assγj

is an infinitesum of equivariant cohomology of the moduli space of semistable representationsof dimensions nγj , n ≥ 0. This is exactly the product decomposition of AV whichwe discussed above for general 3CY categories.

7.4. Donaldson 4d theory, Borcherds automorphic forms. We sug-gested in [KoSo1] that our wall-crossing formulas should be related to those inthe Donaldson theory of 4d manifolds with b+2 = 1 (cf. with the approach via vir-tual fundamental class in [Moch]) as well as with Borcherds hyperbolic Kac-Moodyalgebras and multiplicative automorphic forms. Since Donaldson theory dependson a choice of the gauge group, the latter should be somehow encoded in our story.In case of a complex projective surface S the expected relationship should com-bine the well-known description of Seiberg-Witten theory via complex integrablesystems (see [Don]) with the DT-theory developed in [KoSo1] in the case of thelocal 3CY given by the total space of the anticanonical bundle of S. It is also notclear for us how to relate the wall-crossing formulas in Donaldson theory with thetheory of stability data on graded Lie algebras recalled in Section 5. We think it isan interesting problem.

7.5. WKB approximation in complex domain and WCF. There is astriking similarity between our wall-crossing formulas and identities for the Stokesautomorphisms in the theory of WKB asymptotics for a second order operator onP1

(see e.g. [DelDiPh], Section 3). The latter story was highly motivated by the works

of Ecalle, Pham, Voros and others (see e.g. [DelDiPh]). There is an underlyinggraded Lie algebra of “alien derivatives” in the story, which is probably closelyrelated to the motivic Galois group from [KoSo3]. Despite of a certain similaritywith [KoSo1], Section 2.8 as well as with [BrTL] and some parts of [GaMoNe2] thisrelationship is not clear for us.

7.6. 3CY categories, black holes and the split attractor flow. Here weclarify the discussion in [KoSo1], Section 1.5, 1)-3). More details will appear in[KoSo5]. Cf. also with Section 7.2.

Let MCFT be the “moduli space” of unitary N = 2 superconformal field the-ories. It is believed that MCFT � MA ×MB , where for CFTs associated with a3CY manifold X the moduli space MA is the space fo complexified Kahler struc-tures on X while MB is the moduli space of complex structures on X. Recallthat superstring theory predicts an existence of a family of “compactifying to 10-dimensions” of superconformal field theories parametrized by the 4-dimensionalspace-time endowed with Einstein metric. The latter has singularities at blackholes. Using the time invariance we obtain a metric g on R3 \ {x1, ..., xn}, wherexi are positions of black holes. This family can be interpreted as a harmonic maph : R3 \ {x1, ..., xn} → MCFT .

Let us assume that our CFT is of geometric origin and comes from a 3CYmanifold X. Assume that the Kahler component of h is constant. Then accord-ing to [De1], [DeGrRa] (see aslo [DeMo]) the set of pairs (h, g) is in one-to-onecorrespondence with the set of maps

85


φ : R3 \ {x1, ..., xn} → MX

(here MX is the moduli space of complex structures on X) coming from the fol-lowing ansatz. Namely, the map φ is obtained by the projectivization of the map

φ : R3 \{x1, ..., xn} → Λ, where R3 \{x1, ..., xn} is endowed with the flat Euclideanmetric and Λ is the Lagrangian cone of the moduli space of deformations of Xendowed with a holomorphic volume form (it is locally embedded into H3(X,C)via the period map). We endow Λ with an integral affine structure via the lo-cal homeomorphism Im : Λ → H3(X,R), (τ,Ω3,0

τ ) → Im(Ω3,0τ ), where τ ∈ MX

is a complex structure on X, and Ω3,0τ is the corresponding holomorphic volume

form. Then the ansatz comes from harmonic maps φ which are locally of the form

Im ◦ φ(x) =∑

1≤i≤nγi

|x−xi| + v∞, where γi, 1 ≤ i ≤ n are elements of the charge

lattice Γ = H3(X,Z) (their meanings are the charges of black holes) and v∞ is theboundary condition “at infinity” (see [De1]). This gives us φ.

The image of φ is an “amoeba-shaped” 3-dimensional domain in MX . Conjec-turally, connected components of the moduli space of maps φ with given v∞, γi, 1 ≤i ≤ n are in one-to-one correspondence with split attractor trees (see [De1]).The edges of such a tree are the gradient trajectories of the function |Z(γ)| =

|∫γΩ(3,0)|2/|

∫XΩ(3,0) ∧ Ω(3,0)| . Any edge is locally a projection of an affine line

in Λ with the slope γ ∈ Γ = H3(X,Z). The condition “at infinity”, v∞ ∈ Λ isnormalized so that vol(v∞) = 1. If the split attractor flow (lifted from MX toΛ) starting at v∞ in the direction γ hits the wall of marginal stability so thatγ = γ1 + γ2 + .. + γk then all γ1, ..., γk belong (generically) to a two-dimensionalplane.

One can argue (see more in [KoSo5]) that using our wall-crossing formulas itis possible to find all Ω(γ) := Ω(b, γ), b ∈ Λ, γ ∈ Γ starting with a collection ofintegers Ω(bγ , γ) at the “attractor points” bγ ∈ Λ given by the equation Im bγ = γ.The points C∗bγ ∈ MX are external vertices of the split attractor trees. The wall-crossing formulas are used at the internal vertices of the trees for the computation ofΩ(b, γ). The numbers Ω(bγ , γ) can be arbitrary. We expect that they are determineduniquely by the geometry of certain quaternion Kahler manifold (hypermultipletmoduli space).

For a local 3CY manifold the quaternion Kahler structure reduces to a hy-perkahler one. In this case there is an alternative description of the numbers Ω(γ)as the virtual numbers of holomorphic discs with the boundary on Lagrangian tori(see [KoSo4], [GaMoNe1,2]). This leads to a question about the relationship ofsemistable objects of the A∞-category associated with our local Calabi-Yau mani-fold and holomorphic curves in the hyperkahler manifold. Counting in both casesgives the same numbers.

More generally, one can speculate that for a non-compact quaternion Kahlermanifold with certain behavior “at infinity” one can construct a 3CY categorywhose objects are “quaternion curves” in the twistor space.

7.7. Stability conditions on curves and moduli of abelian differen-tials. Geometry similar to the one on the space of stability conditions appears inthe theory of moduli spaces of holomorphic abelian differentials (see e.g. [Zo]). Themoduli space of abelian differentials is a complex manifold, divided by real “walls”of codimension one into pieces glued from convex cones. It also carries a natural

86


non-holomorphic action of the group GL+(2,R). There is an analog of the centralcharge Z in the story. It is given by the integral of an abelian differential over a pathbetween marked points in a complex curve. We conjectured in [KoSo1] that themoduli space of abelian differentials associated with a complex curve with markedpoints, is isomorphic to the moduli space of stability structures on the Fukaya cat-egory of this curve. As a byproduct one can obtain an example of a non-connectedmoduli space of stability conditions (cf. with [KoZo]).

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[BeFa] K. Behrend, B. Fantechi, Symmetric obstruction theories and Hilbert schemes of pointson threefolds, arXiv:math/0512556.

[BiMi] E. Bierstone, P. Milman, Functoriality in resolution of singularities, ArXiv:math/0702375.

[BonVdB] A. Bondal, M. Van den Bergh, Generators and representability of functors in commu-tative and noncommutative geometry, arXiv:math/0204218.

[Br1] T. Bridgeland, Stability conditions on triangulated categories, arXiv:math/0212237.

[BrTL] T. Bridgeland, V. Toledano Laredo, Stability conditions and Stokes factors,arXiv:0801.3974.

[CeVa] S. Cecotti, C. Vafa, On Classification of N=2 Supersymmetric Theories, arXiv:hep-th/9211097.

[De1] F.Denef, Supergravity flows and D-brane stability, arXiv:hep-th/0005049.

[DeMo] F. Denef, G. Moore, Split States, Entropy Enigmas, Holes and Halos, arXiv:hep-th/0702146.

[DeGrRa] F. Denef, B. Greene, M. Raugas, Split attractor flows and the spectrum of BPS D-branes on the Quintic, arXiv:hep-th/0101135.

[DelDiPh] E. Delabaere, H. Dillinger, F. Pham, Resurgence de Voros et peridodes des courbeshyperelliptiques, An. Inst. Fourier, 43:1, 163-199, 1993.

[DenLo] J. Denef, F. Loeser, Geometry on arc spaces of algebraic varieties, arXiv:math/0006050.

[Don] R. Donagi, Seiberg-Witten integrable systems, arXiv:alg-geom/9705010.

[DonMar] R. Donagi, E. Markman, Cubics, integrable systems, and Calabi-Yau threefolds,

arXiv:alg-geom/9408004.

[GaMoNe1] D. Gaiotto, G. Moore, A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, arXiv:0807.4723.

[GaMoNe2] D. Gaiotto, G. Moore, A. Neitzke, Wall-crossing, Hitchin Systems, and the WKBApproximation, arXiv:0907.3987.

[GuDi] S. Gukov, T. Dimofte, Refined, Motivic, and Quantum, arXiv:0904.1420.

[HaMo] J. A. Harvey, G. Moore, Algebras, BPS States, and Strings, arXiv:hep-th/9510182.

[Jaf] D. Jafferis, Topological Quiver Matrix Models and Quantum Foam, arXiv:0705.2250.

[JafMo] D. Jafferis, G. Moore, Wall crossing in local Calabi Yau manifolds, arXiv:0810.4909.

[Jo1] D. Joyce, Configurations in abelian categories. I. Basic properties and moduli stacks,arXiv:math/0312190.

[Jo2] D. Joyce, Configurations in abelian categories. II. Moduli stacks, arXiv:math/0312192.

[Jo3] D. Joyce, Constructible functions on Artin stacks, arXiv:math/0403305.

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[Jo4] D. Joyce, Configurations in abelian categories. IV. Invariants and changing stability condi-tions, arXiv:math/0410268.

[JoS1] D. Joyce, Y. Song, A theory of generalized Donaldson-Thomas invariants, to appear.

[JoS2] D. Joyce, Y. Song, A theory of generalized Donaldson-Thomas invariants, arXiv:0810.5645

[Ke1] B. Keller, Introduction to A-infinity algebras and modules, arXiv:math/9910179.

[KoSo1] M. Kontsevich, Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariantsand cluster transformations,arXiv:0811.2435.

[KoSo2] M. Kontsevich, Y. Soibelman, Notes on A-infinity algebras, A-infinity categories andnon-commutative geometry. I, arXiv:math/0606241.

[KoSo3] M. Kontsevich, Y. Soibelman, Cohomological Hall algebra and motivic Donaldson-Thomas invariants, to appear.

[KoSo4] M. Kontsevich, Y. Soibelman, Affine structures and non-archimedean analytic spaces,arXiv:math/0406564.

[KoSo5] M. Kontsevich, Y. Soibelman, Split attractor flow, BPS invariants and affine geometry,work in progress.

[KoZo] M. Kontsevich, A. Zorich, Connected components of the moduli spaces of Abelian differ-entials with prescribed singularities, arXiv:math/0201292.

[Moch] T. Mochizuki, Donaldson type invariants for algebraic surfaces. Lecture Notes in Mathe-matics vol. 1972, Springer, 2009.

[Nag1] K. Nagao, Derived categories of small toric Calabi-Yau 3-folds and counting invariants,arXiv:0809.2994.

[Nag2] K. Nagao, Refined open noncommutative Donaldson-Thomas invariants for small crepantresolutions, arXiv:0907.3784.

[NagNak] K. Nagao, H. Nakajima, Counting invariant of perverse coherent sheaves and its wall-crossing, arXiv:0809.2992.

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[OoYa] H. Ooguri, M. Yamazaki, Crystal Melting and Toric Calabi-Yau Manifolds,arXiv:0811.2801.

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M.K.: IHES, 35 route de Chartres, F-91440, France


Y.S.: Department of Mathematics, KSU, Manhattan, KS 66506, USA


89



On the structure of supersymmetricT 3 fibrations

David R. Morrison

Abstract. We formulate some precise conjectures concerning the existenceand structure of supersymmetric T 3 fibrations of Calabi–Yau threefolds, anddescribe how these conjectural fibrations would give rise to the Strominger–Yau–Zaslow version of mirror symmetry.

Mirror symmetry between Calabi–Yau manifolds remains, some twenty yearsafter its discovery, one of the biggest mysteries in mathematics. Originally for-mulated as a physical relationship between certain pairs of Calabi–Yau manifolds[15, 26, 5] (with astonishing mathematical consequences relating enumerative ge-ometry to Hodge theory [14, 61]), the mirror symmetry proposal was refined in1996 by Strominger, Yau, and Zaslow [78] into a much more geometric statement,again based on physics. But while the main idea of the Strominger–Yau–Zaslowproposal has been clear from the outset, many of the details have remained elusive.One of the reasons for this is that the proposal involves special Lagrangian subman-ifolds of a Calabi–Yau manifold, and very few tools are available for studying suchsubmanifolds. As a consequence, the initial period of intense study of the originalStrominger–Yau–Zaslow proposal has largely ended,1 and much of the recent workon mirror symmetry has shifted to other approaches (including reformulations ofthe Strominger–Yau–Zaslow proposal), as is recounted in detail elsewhere in thisvolume.

Our purpose in this paper is to give a quite precise conjectural formulation ofthe Strominger–Yau–Zaslow version of mirror symmetry for Calabi–Yau threefolds.The conjectures we formulate are modifications of conjectures previously made byGross, Ruan, and Joyce; we also restate some other conjectures from [78, 41,55]. Our formulation is unfortunately not directly based on examples, since—as mentioned above—tools for constructing concrete examples are not currentlyavailable. The conjectures are, however, motivated on the one hand by qualitativefeatures of the Strominger–Yau–Zaslow proposal which have been discovered by

2000 Mathematics Subject Classification. Primary 14J32.This research was supported in part by the National Science Foundation (grant DMS-

0606578). Any opinions, findings, and conclusions or recommendations expressed in this materialare those of the author and do not necessarily reflect the views of the National Science Foundation.

1A very readable summary of the progress made on the original proposal, and the transitionto the more recent approaches, was given recently by Gross [34].


1



91

2 DAVID R. MORRISON

mathematicians, and on the other hand by some suggestive arguments from physics.We focus on the motivation from mathematics in this paper.

For simplicity, we restrict our attention for the most part to Calabi–Yau man-ifolds of complex dimension one, two, or three. We expect, however, that similarconjectures could be formulated in higher dimension, at the expense of greatercombinatorial complexity.

1. Supersymmetric torus fibrations

A Calabi–Yau metric is a Riemannian metric on a manifold X of dimension2n whose Riemannian holonomy is precisely SU(n).2 The representation theoryof the holonomy group gives rise to various geometric structures on X: there is acompatible complex structure (unique up to complex conjugation when n ≥ 3), a2-form ω which serves as the Kahler form of the given metric with respect to thatcomplex structure, and a nowhere-vanishing holomorphic n-form Ω. Both ω and Ωare covariantly constant (as is the almost-complex structure operator J ).

Thanks to a result conjectured by Calabi [11] and proven by Yau [80], when Xis a compact Kahler manifold with a nowhere-vanishing holomorphic n-form, thereis a unique Ricci-flat metric in each de Rham cohomology class [ω] ∈ H2(X,R)containing a Kahler form. These metrics have holonomy contained in SU(n), sothey will be Calabi–Yau under our definition provided that the holonomy doesnot reduce to a subgroup. As a consequence, the existence of a nowhere-vanishingholomorphic n-form on a Kahler manifold X is often taken as a definition of Calabi–Yau manifold.

Given a Calabi–Yau metric on X, an associated Kahler form ω, and a holo-morphic n-form Ω, we say that a submanifold L of (real) dimension n is specialLagrangian if ω|L ≡ 0 and Im(eiθΩ)|L ≡ 0 for some θ called the phase of L. Thisnotion was introduced by Harvey and Lawson [45] as a key example of a calibratedgeometry: such submanifolds have a local volume-minimizing property. Unfortu-nately, very few examples of special Lagrangian submanifolds are known in thecompact case.

Let X be a compact Calabi–Yau manifold. Physical arguments predict thatfor most such X, there should exist a mirror partner Y , which is another compactCalabi–Yau manifold whose physical (but not geometrical) properties are closelyrelated to those of X.3 However, as has been recognized since the early days ofmirror symmetry [19, 13, 48, 4], X is expected to have a mirror partner onlywhen the complex structure of X is sufficiently close to a “large complex struc-ture limit point,” which is a class of boundary points in the compactified modulispace MX characterized by having maximally unipotent monodromy [61, 21, 62].Strominger, Yau, and Zaslow [78] argued on physical grounds that any such Calabi–Yau manifold should have a map π : X → B whose general fiber π−1(b) is a specialLangrangian n-torus Tn; such a structure is called a supersymmetric torus fibra-tion of X, since the special Lagrangian condition is the geometric counterpart tothe preservation of half of the supersymmetry in a physical model. Strominger,Yau, and Zaslow also proposed that the mirror partner should be given, to firstapproximation, by a compactification of the family of dual tori

⋃(π−1(b))∨.

2Some authors use the term Calabi–Yau metric when the holonomy is any subgroup of SU(n).3For a brief review of Calabi–Yau manifolds and mirror symmetry, see [66]. More extensive

reviews can be found in [20, 35, 47].

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ON THE STRUCTURE OF SUPERSYMMETRIC T 3 FIBRATIONS 3

It is worth saying a few words about the physical construction of the mirrorpartner. Strominger, Yau, and Zaslow argue that the original Calabi–Yau manifoldand its mirror partner should be related by a physical construction known as “T-duality on the n-torus fibers.” In the absence of holomorphic disks with boundarieson the n-tori, this T-duality simply replaces each nonsingular torus by its dual torus(cf. [58]), while doing something unknown at the singular fibers. This description isexpected to be modified when holomorphic disks are present, but the precise effect ofthe holomorphic disks has not yet been worked out. And as we shall see, the currentexpectation (at least when n = 3) is that such holomorphic disks will be presentfor at least some of the tori in the torus fibration. This has made it difficult toformulate a mathematical version of the original Strominger–Yau–Zaslow proposalwhich is both precise and accurate.

Because the arguments used by Strominger, Yau, and Zaslow implicitly as-sume that the Calabi–Yau metric is uniformly large, we put that hypothesis in thefollowing version of their existence conjecture.

Conjecture 1 (Existence; cf. [78]). For any Calabi–Yau metric on a compactcomplex manifold X of complex dimension n whose complex structure is sufficientlyclose to a large complex structure limit point and whose Kahler class is sufficientlydeep in the Kahler cone, there exists a supersymmetric torus fibration π : X → B,where B is a homology n-sphere.

There is by now considerable indirect evidence in favor of this conjecture, in-cluding an explicit construction in a (slightly degenerate) limiting case [40], as wellas two strategies ([69, 70, 71, 72] and [75, 76, 77]) for constructing weak formsof these fibrations for a certain class of compact Calabi–Yau threefolds. However,it has become clear that proving this conjecture will require developing new tech-niques for studying special Lagrangian submanifolds of a Calabi–Yau manifold. Inspite of our lack of tools to prove the conjecture, though, many qualitative featuresof supersymmetric torus fibrations have been inferred in various ways, and this pa-per is devoted to explaining our best current (conjectural) understanding of thosequalitative features.

We introduce the following terminology and notation. Given a supersymmetrictorus fibration π : X → B, we let Σ ⊂ X be the set of singular points of fibers ofπ, and let Δ = π(Σ) be the discriminant locus of the fibration.

2. Examples in Low Dimension

In low dimension, supersymmetric torus fibrations of compact Calabi–Yau man-ifolds are completely understood.

In the case of elliptic curves (n = 1), any Calabi–Yau metric is flat and a specialLagrangian 1-torus is just a closed geodesic. As is well known, if the homology classis fixed then there is a fibration of the elliptic curve over B = S1 by closed geodesicsin the specified class, with no singular fibers.

In the case of K3 surfaces (n = 2), an analysis is possible due to the non-uniqueness of the compatible complex structure. In fact, the original paper ofHarvey and Lawson [45] showed that if L ⊂ X is special Lagrangian, then thereis a different complex structure on X (compatible with the given Calabi–Yau met-ric) such that L ⊂ X is a complex submanifold. Thus, a special Lagrangian T 2

fibration can be interpreted in another complex structure as a holomorphic elliptic

93

4 DAVID R. MORRISON

fibration π : X → B (with base B = CP1 = S2), and the structure of these is

known in detail. (In fact, a generic Ricci-flat metric on a K3 surface admits sucha fibration [64], so the existence conjecture holds in this case.) Thanks to work ofKodaira [54], a complete classification of possible singular fibers of such fibrationsis known: they are characterized by the conjugacy class of the monodromy actionon H1(T 2,Z). The simplest fibers, called semistable, are associated to unipotentmonodromy transformations. In an appropriate basis, the monodromy matrix takesthe form

(2.1) M =

(1 k0 1

).

The topology of a semistable degeneration with k = 1 is very familiar. One ofthe cycles on the two-torus extends over the degeneration, and the other “vanish-ing” cycle shrinks to a point; if we follow the torus around a loop encircling thedegeneration point in the base, there is a Dehn twist along the vanishing cycle. Inspite of this twisting of the topology of the torus, though, the total space of thefibration is non-singular. (In the physics literature, the corresponding geometry isknown as the “Taub-NUT metric.”)

For the generic elliptic fibration of a K3 surface, all fibers are semistable, andthere are exactly 24 of them, each with k = 1. The monodromy data for such ageneric fibration (choosing an arbitrary base point b ∈ B) gives a natural homo-morphism

(2.2) π1(B − {P1, . . . , P24}, b) → SL(2,Z)

whose generating loops all map to matrices conjugate to eq. (2.1).The mirror partner of a given K3 surface (with a fixed Ricci-flat metric) is

known to be another K3 surface with a different Ricci-flat metric [6]. When passingto the mirror, the monodromy matrices M are replaced by tM−1; since tM−1 isconjugate to M , the monodromy data does not change. We will conjecturallyextend this kind of “topological” mirror symmetry statement to dimension 3 in thenext section. Note that, in any dimension, if we replace all nonsingular tori by theirdual tori, the monodromy matrices change as M �→ tM−1.

3. Smooth T 3 fibrations

The first step in studying supersymmetric T 3 fibrations of compact Calabi–Yauthreefolds is to study more general T 3 fibrations, without imposing the “special La-grangian” condition. For Calabi–Yau hypersurfaces in toric varieties (of arbitrarydimension), Zharkov [81] constructed a (topological) Tn fibration. A general pro-gram to understand such fibrations π : X → B (in dimension 3) for which the mapπ is smooth (i.e., C∞) was initiated by Gross [28, 29, 31], and parallel results wereobtained by Ruan [69, 70, 71] in his study of a specific class of T 3 fibrations ofCalabi–Yau hypersurfaces. The monodromy of such fibrations and the topology ofthe singular fibers was determined under a suitable assumption of genericity, anal-ogous to the assumption of “generic elliptic fibration” in the case of K3 surfaceswhich guaranteed that all fibers were semistable with k = 1. We can summarizethe analysis in a general conjecture (which conjecturally extends their results tothe general case).

94


Conjecture 2 (Topology; cf. [28, 29, 31, 69, 70, 71]). Let π : X → Bbe a smooth T 3 fibration4 of a compact Calabi–Yau threefold which is generic in asuitable sense. Then

a) The discriminant locus of the fibration is a trivalent graph Γ.b) The topology near the edges of Γ is modeled by the product of a cylinder

with the k = 1 semistable degeneration of two-tori. In particular, a Dehntwist along the vanishing cycle and a nonsingular total space are featuresof this topology.

c) For any loop around an edge of Γ, the monodromy on either H1 ∼= H2 orH1 of the 3-tori is conjugate to

(3.1) M =

⎛⎝

1 0 10 1 00 0 1

⎞⎠ .

In particular, both monodromy actions have a 2-dimensional fixed plane.d) The vertices of Γ come in two types: near a positive vertex, the three

monodromy actions on H1 ∼= H2 near the vertex have fixed planes whoseintersection is 1-dimensional, while the three monodromy actions on H1

have a common 2-dimensional fixed plane. In an appropriate basis, themonodromy matrices on H1 take the form

(3.2)

⎛⎝

1 0 10 1 00 0 1

⎞⎠ ,

⎛⎝

1 0 00 1 10 0 1

⎞⎠ ,

⎛⎝

1 0 −10 1 −10 0 1

⎞⎠ .

On the other hand, near a negative vertex, the three monodromy actionson H1 ∼= H2 near the vertex have a common 2-dimensional fixed plane,while the three monodromy actions on H1 have fixed planes whose intersec-tion is 1-dimensional. In an appropriate basis, the monodromy matriceson H1 take the form

(3.3)

⎛⎝

1 0 10 1 00 0 1

⎞⎠ ,

⎛⎝

1 1 00 1 00 0 1

⎞⎠ ,

⎛⎝

1 −1 −10 1 00 0 1

⎞⎠ .

e) The fiber of π over any point of B other than a vertex of Γ has a fixedpoint free U(1) action and in particular has Euler characteristic 0. Thefiber of π over each positive vertex has Euler characteristic 1, and the fiberover each negative vertex has Euler characteristic −1. (In fact, Gross [31]and Ruan [71] gave explicit descriptions of these singular fibers, but wewill not reproduce those descriptions here.)

There is an induced global monodromy action on H1

(3.4) π1(B −Δ, b) → SL(3,Z),

whose generators satisfy the conditions spelled out in the conjecture. If we have acompact Calabi–Yau threefold and its mirror partner, with smooth T 3 fibrationswhose nonsingular fibers are dual to each other, then the monodromy transforma-tions will be related as M �→ tM−1. This implies that the roles of the positiveand negative vertices in the fibration are reversed between a Calabi–Yau threefold

4We stress that it is the map π which is smooth, not the fibers of the fibration.

95

6 DAVID R. MORRISON

Figure 1. A triangulation of a two-dimensional face of the New-ton polytope of the quintic threefold, and the corresponding dualgraph.

and its mirror partner. Since part (e) of the conjecture implies that the topologicalEuler number of X can be calculated via

(3.5) χtop(X) = #{positive vertices} −#{negative vertices},the effect of mirror symmetry on monodromy then shows that the Euler numberchanges sign:

(3.6) χtop(Y ) = #{negative vertices} −#{positive vertices} = −χtop(X),

as expected from physical mirror symmetry arguments.In [31], Gross showed how to go further, and use the data from a generic smooth

T 3 fibration of a given compact Calabi–Yau threefold to construct a manifold whichis a candidate mirror partner. The transpose inverse of the original monodromyrepresentation produces a mirror monodromy representation, describing the mon-odromy on the family of dual tori (with positive and negative vertices reversed).Gross proved a “Reconstruction Theorem:” the family of dual tori can be com-pleted to a compact topological manifold with a smooth T 3 fibration, satisfying theproperties stated in the conjecture.

4. Combinatorics of Γ

For smooth T 3 fibrations of a compact Calabi–Yau threefold, the combinatoricsof the graph Γ are beautiful and intricate. For example, in the case of a quintichypersurface in CP

3, the graph depends on choices of triangulations of the two-dimensional faces of the Newton polytope of the defining equation; one such choiceis shown on the left side of Figure 1. One constructs the dual graph of each suchtriangulation, in which each face of the triangulation gives a vertex of the graph,and each edge of the triangulation is crossed by an edge of the graph (as shown onthe right side of Figure 1). That dual graph then becomes a piece of Γ (illustratedon the left side of Figure 2) in which each vertex is “negative.” The pieces areassembled according to the combinatorics of the Newton polytope, in which thefree ends of the dual graph meet free ends from other faces of the Newton polytope,forming trivalent vertices which are the “positive” vertices of Γ. The pieces thusattach three at a time; a neighborhood of one such attachment is illustrated on theright side of Figure 2

96


Figure 2. Portions of Γ for the quintic threefold.

A general description of these graphs, for Calabi–Yau complete intersections intoric varieties, was given by Haase and Zharkov [42, 43, 44] and by Gross [33].Their results5 show that the classes of mirror pairs described by Batyrev [8] and byBatyrev and Borisov [10, 9] admit smooth T 3 fibrations satisfying Conjecture 2,and that those T 3 fibrations are mirror duals of each other.

This theory of smooth T 3 fibrations of compact Calabi–Yau threefolds canalso be related to some topological aspects of mirror symmetry which have playedimportant roles in the physics literature [79, 4, 25]. First, the construction of Γ fora Calabi–Yau hypersurfaceX in a toric fourfold depends on a choice of triangulationof the faces of the Newton polytope, and this choice of triangulation is equivalent toa choice of large complex structure limit point in the moduli space [23, 4]. Mirrorsymmetry (as developed in the physics literature) offers an alternate interpretation:each large complex structure limit point corresponds to a different birational modelof the mirror partner, and the Kahler cones of the birational models fit together intoa common space (after complexification), mirroring the complex structure modulispace MX [63]. In this interpretation, the choice of birational model dependsexplicitly on a choice of triangulation of the Newton polytope (which describesthe mirror toric fourfold in Batyrev’s construction [8]). The simplest birationalchange—a “flop”—is realized by the simple change of triangulation illustrated inFigure 3. As Gross pointed out [31, Remark 4.5], the corresponding change in dualgraph (also illustrated in the figure) has the correct monodromy properties to beallowed as a new graph Γ′. One expects that by appropriately varying the complexstructure and the Kahler metric, the connecting edge in the original graph willshrink to zero length, giving the 4-valent vertex shown in the intermediate stage;further variation then causes a new connecting edge to grow, changing the topologyto that of Γ′.

The intermediate step illustrated in the middle of Figure 3 is a partial trian-gulation, which is expected to correspond to a “conifold” singularity on the mirrorCalabi–Yau threefold. A second topological feature of mirror symmetry is the“conifold transition” [12, 25] in which that conifold singularity is resolved witha small blowup, rather than being smoothed with a change of complex structure.(The conifold singularity is on the mirror partner, but this transition can also be

5The proofs of some of the results stated by Gross [33] were deferred to another paper whichhas not yet appeared.

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8 DAVID R. MORRISON

Figure 3. The change of triangulation corresponding to a flop.

described on the original Calabi–Yau threefold [65].) The graph Γ appears tochange as follows, as proposed independently by Gross [30] and Ruan [73]: aftershrinking the connecting edge to zero size, leaving a 4-valent vertex, the two armsof the graph crossing at that vertex are separated into different planes, as illus-trated in Figure 4. As Gross and Ruan verify, this change is compatible with themonodromies around the edges and produces the expected change in topologicalEuler characteristic for a conifold transition. However, the relation between thisconstruction and more global versions of the conifold transition remains mysteriousand needs further study.

5. Affine structures on the base

Local moduli for a compact special Lagrangian submanifold L of a compactCalabi–Yau manifold X were determined by McLean [59]: the deformation spaceis smooth, and its tangent space is canonically identified with the space of harmonic1-forms of L. Hitchin [46] used this identification to construct two affine structureson B −Δ (if π is smooth), which geometrize the monodromy transformations thatoccurred in Conjecture 2.

If V is a normal vector field to L in X, then the contraction ι(V )ω of V withthe Kahler form gives a harmonic 1-form on L, and the contraction ι(V )Ω of V withthe holomorphic n-form gives a harmonic (n − 1)-form on L. These constructionsgive rise to the affine structures, which can be seen by considering periods of theharmonic forms. If ∂

∂ti, . . . , ∂

∂tnare vector fields on the deformation space M which

span the tangent space to M at [L], and A1, . . . , An is a basis for H1(L,Z), thenwe can form the period matrix

(5.1) λij =

∫Ai

ι(∂

∂tj)ω.

The 1-forms∑

j λijdtj on the deformation space M are closed, and can be inte-

grated to give local coordinates u1, . . . , un on M at [L] satisfying dui =∑

j λijdtj .Such coordinate systems provide an affine structure, that is, the transition func-tions between any two such coordinate systems lie in the affine group Rn�GL(Zn).Intrinsically, the lattice Zn should be identified with H1(L,Z) in this case, and thisaffine structure carries the information about the monodromy on H1(L,Z).

Similarly, if B1, . . . , Bn is a basis for Hn−1(L,Z), then we can form the periodmatrix

(5.2) μij =

∫Bi

ι(∂

∂tj)Ω.

98


Figure 4. Expected change of graph for a conifold transition.

The 1-forms∑

j μijdtj on the deformation space M of L are closed, and can be in-

tegrated to give local coordinates v1, . . . , vn on M satisfying dvi =∑

j μijdtj . Suchcoordinate systems provide the other affine structure, which carries the informationabout the monodromy on Hn−1(L,Z) ∼= H1(L,Z).

Hitchin shows that these two affine structures are related by a Legendre trans-form with respect to a suitable locally defined function K (which also determines acanonical metric on the deformation space). This is a version of mirror symmetrywhich is formulated strictly on the base of the torus fibrations, a notion which wasfurther developed in [41, 55].

6. The large complex structure limit

When X is a compact Calabi–Yau threefold, it is expected that the map πgiving a supersymmetric torus fibration π : X → B will be only piecewise smooth,so the analysis of Sections 3 and 5 does not directly apply. Gross showed [29] thatif π is smooth, the discriminant locus Δ ⊂ B must have codimension two; in themore general “piecewise smooth” case, Δ may have codimension 1. However, wedo expect that Δ will always have a retraction onto a subset Γ of codimension two.And there is a particular limiting situation in which this retraction should becomeevident: the large complex structure limit.

In fact, Gross–Wilson [41] and Kontsevich–Soibelman [55] have formulateda precise conjecture about the large complex structure limit of a supersymmetrictorus fibration.6

Conjecture 3 (Large Complex Structure Limit; cf. [41, 55]). Let X → S bea maximally unipotent degeneration of compact simply-connected Calabi–Yau man-ifolds of complex dimension n, degenerating at 0 ∈ S, let si ∈ S be a sequencewith lim si = 0, and let gi be a sequence of Ricci-flat metrics on Xsi with diameterbounded above and below. Then there exists a subsequence (Xsij

, gij ) which con-

verges in the sense of Gromov–Hausdorff [27] to a metric space (X∞, d∞), whereX∞ is homeomorphic to to the sphere Sn, and d∞ is induced by a Riemannianmetric on X∞ − Γ∞ for some Γ∞ ⊂ X∞ of codimension two.

Following the affine structure to this limit, one expects to find a limiting affinestructure, and in fact the discriminants Δik should have collapsed to Γ∞ in thelimit.

6Note there has been substantial additional progress on these limits and on various structureson the base in subsequent work of Kontsevich and Soibelman [56, 57], reviewed elsewhere in thisvolume.

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10 DAVID R. MORRISON

Figure 5. The discriminant locus in the (x2, x3)-plane for theHarvey–Lawson fibration of C3.

7. Non-compact examples of special Lagrangian fibrations

Harvey and Lawson’s original paper about calibrations [45] gave an explicitexample of a special Lagrangian fibration.7 Define f : C3 → R3 by

(7.1) f(z1, z2, z3) = (Im(z1z2z3), |z1|2 − |z2|2, |z1|2 − |z3|2).Then the fibers of f are special Lagrangian, and are all invariant under the actionof the diagonal torus with determinant 1:

(7.2) {diag(eiθ1 , eiθ2 , eiθ3) | θ1 + θ2 + θ3 = 0}.The singularities of fibers are located where zi = zj = 0 for some pair of indices

i and j; the images of these give three rays within the plane {x1 = 0} ⊂ R3, namely(i) x2 = 0, x3 ≤ 0, (ii) x2 ≤ 0, x3 = 0, and (iii) x2 = x3 ≥ 0. This is illustrated inFigure 5.

The nonsingular fibers are all homeomorphic to T 2 × R. Note that when-ever x1 = 0, if we write zj = rje

iθj then either θ1 + θ2 + θ3 = 0 (in which caseRe(z1z2z3) ≥ 0), or θ1 + θ2 + θ3 = π (in which case Re(z1z2z3) ≤ 0). Thus, thereare two natural subsets f−1(0, x2, x3)

± of the fiber f−1(0, x2, x3), distinguished bythe sign of Re(z1z2z3). These subsets meet along

(7.3) f−1(0, x2, x3) ∩ {z1z2z3 = 0}.When the fiber is smooth, each subset is a manifold with boundary, and they

meet along their common boundary. However, when the fiber is singular and(x2, x3) �= (0, 0), each subset is itself a smooth special Lagrangian submanifold,homeomorphic to S1 × R2. Note that f−1(0, 0, 0) is a special case: as Harvey andLawson pointed out, each subset f−1(0, 0, 0)± is a cone over T 2, and those conesmeet precisely at the origin in C

3.Joyce [49] builds some new special Lagrangian fibrations by carefully combining

subsets of Harvey–Lawson fibers. Let

(7.4)N+

a = {|z1|2 − a = |z2|2 + a = |z3|2 + |a|,Im(z1z2z3) = 0,Re(z1z2z3) ≥ 0}

and

(7.5)N−

a = {|z1|2 − a = |z2|2 + a = |z3|2 + |a|,Im(z1z2z3) = 0,Re(z1z2z3) ≤ 0}.

7Further development of examples of this type was made by Goldstein [24] and Gross [30],and they have been extensively studied in the physics literature (e.g., in [3, 2, 1]).

100


Figure 6. The discriminant locus as a ribbon.

Then

(7.6) N+a =

{f−1(0, 2a, 2a)+ when a ≥ 0,

f−1(0, a, 0)+ when a ≤ 0.

and similarly for N−a .

To build a special Lagrangian fibration, Joyce considers translations of thesemanifolds for c ∈ C. Let

(7.7)N±

a,c = {|z1|2 − a = |z2|2 + a = |z3 − c|2 + |a|,Im(z1z2(z3 − c)) = 0,±Re(z1z2(z3 − c)) ≥ 0.}

These can be made the fibers of special Lagrangian fibrations by defining F± :C3 → R× C by

(7.8) F±(z1, z2, z3) =

⎧⎪⎨⎪⎩

( 12 (|z1|2 − |z2|2), z3) if z1 = z2 = 0

( 12 (|z1|2 − |z2|2), z3 ∓ z1z2|z1| ) if |z2|2 ≤ |z1|2 �= 0

( 12 (|z1|2 − |z2|2), z3 ∓ z1z2|z2| ) if |z2|2 > |z1|2

.

With this definition, (F±)−1(a, c) = N±a,c.

Notice that the fibrations F± are only piecewise smooth, and that the discrim-inant locus in each case is {(0, c)} ⊂ R× C, which has codimension 1. Notice alsothat when a > 0, both N+

a,c and N−a,c contain the boundary of the holomorphic

disk {|z1|2 ≤ a} of area 2πa, and that when a < 0, both N+a,c and N−

a,c contain the

boundary of the holomorphic disk {|z2|2 ≤ −a} of area −2πa. In some sense, theseshrinking disks are “responsible” for the singularity being created at a = 0.

To go further, Joyce invokes the extensive theory which he developed in [50,51, 52] concerning the structure of special Lagrangian 3-manifolds with a U(1)action. Using that theory, he is able to construct [49, Theorem 6.5] a special

Lagrangian fibration F : C3 → R3 whose discriminant locus is a ribbon, that is, thelocus {(0, x2, x3) | 0 ≤ x2 ≤ 1} ⊂ R3, as illustrated in Figure 6. There are severalimportant properties of this example of Joyce’s. First, the fiber over an interiorpoint of the ribbon has two singularities—one locally modeled by F+ and the otherlocally modeled by F−. Second, as in those local models, there are holomorphicdisks with boundary in the fiber for x1 �= 0 (and 0 < x2 < 1), whose area approaches0 as x1 approaches 0; in fact, there is one such holomorphic disk for each of the twosingular points.

Third, as we approach the boundary of the strip within the plane x1 = 0,something interesting happens: the two bounding circles approach each other andthe holomorphic disks cancel out as the boundary of the strip (either x2 = 0 orx2 = 1) is reached. There are no holomorphic disks when x2 < 0 or x2 > 1. Theregion in which holomorphic disks are present is illustrated in Figure 7.

101


Figure 7. The nearby tori which bound a holomorphic disk.

Thus, along the boundary of the strip, one has a singularity of multiplicity 2(in an appropriate sense), which bifurcates into a pair of singularities in the middleof the strip, and those singularities rejoin at the other boundary.

Note that the plane which contains the discriminant locus can be identified in-trinsically using local affine coordinates. The cycle γ ∈ H1(N

±a,c,Z) which bounds

a holomorphic disk is the vanishing cycle for the family, and it defines a dual sub-space in γ⊥ ⊂ H1

c (N±a,c,Z), which can be locally identified with the plane containing

the discriminant locus. This plane can also be characterized as the monodromy-invariant plane in the compactly-supported cohomology of the fiber.

Joyce conjectures that his examples exhibit generic behavior. In fact, even La-grangian fibrations (not just special Lagrangian fibrations) are expected to exhibitthese phenomena, as explained in [18].

Conjecture 4 (Singular Fibers; cf. [49]). Let π : X → B be a supersymmetricT 3 fibration of a compact Calabi–Yau threefold with respect to a Calabi–Yau metricwhose compatible complex structure is sufficiently close to a large complex structurelimit point, and whose Kahler class is sufficiently deep in the Kahler cone. Then

(1) The discriminant locus Δ ⊂ B has codimension one. In affine coordi-nates, Δ is locally contained in the plane corresponding to the monodromy-invariant subspace of H1(π−1(b),Z) for b near Δ.

(2) The fiber over the general point of Δ has two singular points, one of whichis modeled locally by F+ and the other of which is modeled locally by F−.

(3) The fiber over the general point of the boundary of Δ is modeled locally by

F .(4) Let H ⊂ B be the set of fibers which contain the boundary of at least

one holomorphic disk in X, or are singular. Then the boundary of Δ iscontained in the boundary of H.

The last statement about which fibers contain the boundaries of disks was notconjectured by Joyce, but is consistent with the behavior exhibited by his example

F (as illustrated in Figure 7).

102


-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

Figure 8. Amoeba of z + w + 1.

8. Amoebas

A common feature of the constructions of Zharkov [81] and Ruan [69, 70, 71,72], which dovetails nicely with the analysis of Joyce described in Section 7, isthe description of the discriminant locus of a supersymmetric torus fibration as anamoeba. Amoebas were introduced by Gelfand, Kapranov, and Zelevinsky [23]; wewill briefly review the theory, following Mikhalkin [60] (see also [32]).

Let f =∑

aIxI be a Laurent polynomial in n complex variables. (Here, I is

a multi-index with negative powers allowed, but f has only finitely many non-zeroterms.) The amoeba of f is the set Af = Log(Vf ), where Vf = {z ∈ (C∗)n | f(z) =0}, and Log : (C∗)n → Rn is defined by

(8.1) Log(z1, . . . , zn) = (log |z1|, . . . , log |zn|).

A simple example is given by f(z, w) = z +w+ 1, which is often chosen because itcan be graphed exactly, as in Figure 8. More complicated examples have “holes”in the amoeba, as indicated on the left side of Figure 9.

The Laurent polynomial f has an associated Newton polytope Δf and toricvariety Tf , and there is a moment map μ : Tf → Δf ⊂ R

n once a symplecticstructure has been chosen on Tf . The closure of μ(Vf ) is called the compactifiedamoeba of f . Note that Log and μ are closely related: the interior of the imageΔf of μ is mapped homeomorphically by Log ◦μ−1 to all of Rn. An example of acompactified amoeba is illustrated on the right side of Figure 9.

Forsberg, Passare, and Tsikh [22] showed that each component of the com-plement Rn − Af is convex, and that there is an injective map from the set ofcomponents to the lattice points Δf ∩ Zn in the Newton polytope, in which thebounded components (the “holes”) map to points in the interior of Δf . For a givenpolyhedron Δ, there exist functions f with Δf = Δ whose amoebas have the max-imum number of holes, but typically there also exist functions with Δf = Δ whoseamoebas have fewer holes.

In the case n = 2, a formula of Baker [7, 53] identifies the genus of a smoothcompactification of the affine curve Vf with the number of interior lattice points inthe Newton polytope. Thus, in that case, the maximum number of holes coincideswith the genus. There is an associated topological picture when the number ofholes is maximal: the map Log will in this case be 2-to-1 over the interior of theamoeba, and 1-to-1 on the boundary of the amoeba. It is easy to see that this givesthe right answer for the genus.

103


Figure 9. An amoeba with one hole, and its compactification.

Additional information about the amoeba can be obtained by considering theRonkin function Nf : Rn → R defined by

(8.2) Nf (x1, . . . , xn) =1

(2πi)n

∫μ−1(x1,...,xn)

log |f(z1, . . . , zn)|dz1z1

∧ · · · ∧ dznzn

.

Ronkin [68] and Passare–Rullgard [67] show that Nf is well-defined on all of Rn,is convex over Af , and is locally linear on the complement of Af . Let {E} be theset of components of the complement, and let NE be the extension of Nf |E to alinear function on all of Rn. Passare and Rullgard define

(8.3) N∞f = max

ENE ,

which is a piecewise linear function on Rn, and then define the spine of the amoeba

Af to be the set Sf ⊂ Rn of points at which the function N∞f is not locally linear.

A key theorem of [67] is that the spine Sf is a strong deformation retract of theamoeba Af .

Note that the spine of the amoeba which is shown in Figure 8 is precisely givenby Figure 5.

Ruan [74] observed that for a Calabi–Yau hypersurface X in a toric fourfoldT which is close to the large complex structure limit, the intersections Cjk =X ∩Tj ∩Tk with pairs of toric divisors have amoebas with the maximum number ofholes, and these amoebas retract to their spines as the large complex structure limitis approached. The spines in fact form the pieces of the graph Γ used to describea topological T 3 fibration, which are dual graphs of appropriate triangulations ofthe Newton polytopes (as was illustrated in Figure 1).

An amoeba for Cjk, together with its spine, is shown in the case of the quinticthreefold in Figure 10. The spine is precisely the graph which occurred on the leftside of Figure 2.

As in the case of Γ itself, moving among different large complex structure limitpoints causes the combinatorics of the triangulation to change (as discussed at theend of Section 3); we expect a corresponding change in the combinatorics of theamoebas.

9. Reconstruction in the Lagrangian case

Gross’s reconstruction theorem produced a topological 6-manifold out of thedata describing a smooth T 3 fibration, and this provides (in principle) a method

104


Figure 10. Amoeba and spine for Cjk of the quintic threefold.

for constructing mirror partners when they are not known, provided that one hasa smooth T 3 fibration.

Castano-Bernard and Matessi [18] have proved an analogous theorem whichproduces a compact symplectic 6-manifold with a piecewise smooth Lagrangian T 3

fibration, starting from the data describing the smooth fibers of this fibration.8

The starting data in this case is the affine structure on the base—a refinement ofthe simple monodromy data which Gross’s theorem needed. The key techniqueof stitching together smooth T 3 fibrations along a common boundary had beendeveloped in earlier work of these authors [17, 16].

A particularly interesting feature of Castano-Bernard and Matessi’s construc-tion is the behavior of the discriminant locus Δ. Their fibrations have a discriminantlocus which is a trivalent graph near the positive vertices, but has codimension 1near the negative vertices. The discriminant locus retracts onto a trivalent graph;the inverse “thickening” of parts of this graph to a codimension 1 set replaces eachneighborhood of a negative vertex with an amoeba-like shape which retracts backto the graph. Moreover, the fibration is smooth outside of a set which retracts to (asubset of) the graph. This result, when combined with the discussion in Section 8,helps to motivate our conjectures in the next two Sections.

10. The geometry of T 3 fibrations

Prior to Joyce’s analysis of the structure of special Lagrangian fibrations [49],there had been speculation that supersymmetric T 3 fibrations of compact Calabi–Yau threefolds would always be smooth, so that the detailed structure (in thegeneric case) would be the one given in Conjecture 2. However, Joyce’s analysisprompted many of us to rethink the question, and to try to formulate propertiesanalogous to those of Conjecture 2 which we would expect supersymmetric T 3

fibrations to have. Once such formulation appears in Conjecture 5 below.In Conjecture 2, the discriminant locus is a graph Γ, but in general we should

expect a discriminant locus Δ which only retracts to a graph Γ. As Joyce pointedout (and was already mentioned in Conjecture 4), the first thing to expect is thatthe edges of the graph Γ should thicken to ribbons; moreover, one should see two

8Note that this is still not a special Lagrangian fibration, but provides an important inter-mediate step between the cases of topological fibration and special Lagrangian fibration.

105


Figure 11. The discriminant locus near a negative vertex.

singular points in each fiber over an interior point of the ribbon, with the two pointscoming together to a single singular point along the edges of the ribbon. The nextthing to expect was also proposed by Joyce [49]: since at a “negative” vertex, thelocal monodromy transformations share a common 2-dimensional fixed plane, Δshould remain planar, and the negative vertex should be replaced by a “trivalentribbon” of the sort illustrated in Figure 11. (This is the same structure foundby Castano-Bernard and Matessi [18] in the Lagrangian case.) Near a “positive”vertex, the three planes containing parts of Δ share a common line but are distinct;Joyce also made a specific proposal for the structure in this case, but we will makea slightly different proposal in our main conjecture below.

Another motivation for our conjecture is the observation by Ruan that in hisconstruction (and also in Zharkov’s construction), the discriminant locus is builtout of amoebas, in fact, out of amoebas with the maximum number of holes. Sincesuch amoebas arise from moment maps which are 2-to-1 over the interior and 1-to-1 over the edges, it is natural to identify the set of singular points of π with thealgebraic curve whose moment map image is the amoeba. This is what we do inour main conjecture.

A third input to our conjecture is the Harvey–Lawson fibration of C3, whichis the standard model of a “positive” vertex with a Lagrangian structure. In thatfibration, the set of singular points consists of the coordinate axes in C3, whichmeet in a “transverse triple point.” We conjecture that this is a general propertyof positive vertices.

Conjecture 5 (Geometry). Let π : X → B be a supersymmetric T 3 fibra-tion of a compact Calabi–Yau threefold with respect to a Calabi–Yau metric whosecompatible complex structure is sufficiently close to a large complex structure limitpoint, and whose Kahler class is sufficiently deep in the Kahler cone. Then π ispiecewise smooth and

a) The set Σ ⊂ X of singular points of fibers of π is a complex subvariety ofX of complex dimension 1.

b) All singular points of Σ are transverse triple points, locally of the form{z1z2 = z1z3 = z2z3 = 0} for local complex coordinates z1, z2, z3.

c) For each connected component Σα of Σ, π(Σα) is contained in a (real)surface Aα ⊂ B, and the map π|Σα

is generically 2-to-1 onto its imageπ(Σα), which has the topology of a compactified amoeba with g(Σα) holes.

106


Figure 12. Tori which bound a holomorphic disk: the global picture.

d) The discriminant locus Δ retracts to a trivalent graph Γ which is the unionof the spines of the (topological) compactified amoebas π(Σα). The graphΓ has all of the properties in Conjecture 2.

e) The positive vertices of the graph Γ are the points in Δ at which the spinesof the various compactified amoebas meet. The map π puts the singularpoints of Σ in one-to-one correspondence with the positive vertices

f) The set H ⊂ B of fibers which are either singular or contain the boundaryof at least one holomorphic disk retracts onto the discriminant locus Δ.(Although in the local example illustrated in Figure 7 the set H extendedfar away from Δ, we expect that in global examples H will be confined toa small neighborhood of Δ, as illustrated in Figure 12.) The map π issmooth9 when restricted to X − π−1(H).

We have been deliberately vague about the notion of a topological amoebaand its spine, since we don’t know how much of the theory of amoebas should beexpected to go through. It would be very interesting to know, for example, if someversion of the Ronkin function can be defined for a supersymmetric T 3 fibration.

Note that one of the things which could happen if we attempt to deform thisstructure too far away from a large complex structure limit point is that π|Σα

mightstop being generically 2-to-1, as happens for moment maps of algebraic curves. Itwould be very interesting to see what happens to supersymmetric T 3 fibrations inthat case. Presumably, something more general than Joyce’s phenomenon of twosingular points per fiber is going on here (as Joyce briefly discusses in [49, Section8.2]).

Among the consequences of our conjecture is a specific prediction for the struc-ture of Δ near a positive vertex. Using local affine coordinates to identify a neigh-borhood of the vertex with the first cohomology of the fiber, the three local piecesof Δ must be contained in the three monodromy-invariant 2-planes, which meetalong a common line but are distinct. However, because the corresponding sin-gular point of Σ is a transverse triple point, the thickening of each piece of thediscriminant locus will need to “thin down” near the positive vertex so that thethree pieces of Δ meet in a single point, leading to a description of the discriminantlocus similar to that illustrated in Figure 13. (This “thinning down” was absentfrom Joyce’s proposal about the positive vertices.) Notice that, as Joyce observed,the discriminant locus has a markedly different local structure near positive and

9The referee points out that due to the smoothness of local moduli for special Lagrangiansubmanifolds [59], we should even expect π to be smooth on the larger set X − π−1(Δ).

107


Figure 13. The discriminant locus near a positive vertex.

negative vertices and therefore we cannot hope for a mirror symmetry statementwhich simply dualizes all nonsingular tori in the fibration.

The conjecture that π is smooth on a region whose complement retracts to Δwas motivated in part by the properties of the construction of Castano-Bernard andMatessi [18]. The conjecture that H provides such a region is motivated in part byJoyce’s observation that—at least in examples—the boundary between the set oftori bounding holomorphic disks and the set not bounding holomorphic disks is aboundary along which π fails to be smooth. An additional motivation for part (f)of the conjecture is the hope that a proper understanding of the disk contributionsto the physical “T-duality” construction will restore the symmetry between theT 3 fibrations on the original Calabi–Yau manifold and its mirror partner: thefibrations would consist of dual tori on the complement of H (where Hitchin’sLegendre transform will relate the affine structures), with the duality between thetori somehow modified within H by the disk contributions.

11. Degenerations

We close with a final conjecture, which is perhaps less well-motivated thanConjecture 5, but which proposes an explanation for why the structures we expectfrom supersymmetric T 3 fibrations are related to the complex structure being neara large complex structure limit point (as the physics suggests). This final conjec-ture also points the way towards a connection between our conjectures and theinteresting program of Gross and Siebert [36, 37, 38, 39], which formulates mirrorsymmetry in terms of degenerations of algebraic varieties.

Our final conjecture essentially says that the algebraic curves Σα should arisefrom a large complex structure degeneration of the Calabi–Yau threefolds.

Conjecture 6 (Degeneration). Let X → S be a proper flat family of threefoldswhose generic point Xη is a Calabi–Yau threefold, and whose fiber Xs0 at somespecial point s0 ∈ S is a large complex structure degeneration of the form Xs0 =⋃Xj, where the Xj are the components of Xs0 . Equip X → S with a relative Kahler

metric g whose Kahler classes are sufficiently deep in the Kahler cone. Then thereexist (non-flat) families of subvarieties Cjk ⊂ X such that (Cjk)s0 = Xj ∩Xk, but

108


(Cjk)s is nonsingular of complex dimension 1 when Xs is nonsingular,10 such thatfor all s sufficiently close to s0 there is a supersymmetric T 3 fibration of Xs withrespect to gs whose singular locus Σs is precisely

⋃(Cjk)s.

This is the structure found in the case of Calabi–Yau hypersurfaces in toricfourfolds: in that case, each Xj ∩ Xk is an intersection of toric divisors, whichmeets the nearby nonsingular Calabi–Yau threefolds Xs in a complex curve (Cjk)s;the union of those curves, in the constructions of Zharkov and of Ruan, forms theset Σs of singular points of Xs.

Acknowledgments. It is a pleasure to thank Emanuel Diaconescu, RobbertDijkgraaf, Mark Gross, Christian Haase, Dominic Joyce, Ronen Plesser, CumrunVafa, and Ilia Zharkov for useful conversations about the topics reported on here,and to thank the referee for some helpful comments.

I would also like to thank Kansas State University for hosting the conferenceon Tropical Geometry and Mirror Symmetry, and the Aspen Center for Physicswhere the writing of this paper was completed.

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University of California, Santa Barbara


112


Log Hodge Groups on a ToricCalabi-Yau Degeneration

Helge Ruddat

Abstract. We give a spectral sequence to compute the logarithmic Hodgegroups on a hypersurface type toric log Calabi-Yau space X, compute its E1

term explicitly in terms of tropical degeneration data and Jacobian rings andprove its degeneration at E2 under mild assumptions. We prove the basechangeof the affine Hodge groups and deduce it for the logarithmic Hodge groups inlow dimensions. As an application, we prove a mirror symmetry duality indimension two and four involving the ordinary Hodge numbers, the stringyHodge numbers and the affine Hodge numbers.

Introduction

Hodge theory implies that Hodge numbers stay constant in smooth, properfamilies [11]. By using logarithmic differential forms Steenbrink extended thisresult to normal crossing degenerations [32]. Later it was observed [26], [29] thatthe notion of log smoothness in abstract log geometry [24],[25] provides the rightframework for this kind of result.

In [19] and [21] Gross and Siebert provide a framework for a comprehensive un-derstanding of mirror symmetry via maximal degenerations X → S, using the tech-nique of log geometry. The central fibre of their maximal degenerations are unionsof complete toric varieties, and they allow an essentially combinatorial (“tropical”)description via an integral affine manifold B with certain singularities along witha compatible decomposition into lattice polyhedra. While a maximal degenerationdoes not literally define a log smooth morphism, it is shown in [20] that in manycases there is enough log smoothness to compute the Hodge numbers of the generalfibers from the log Hodge numbers of the central fiber. The latter can in turn becomputed on B:

hp,q(Xt) = hp,qaff (B) := hq(B,

p∧i∗Λ⊗Z R).

1991 Mathematics Subject Classification. Primary 14J33, 32S35; Secondary 14M25, 52B70.Key words and phrases. Algebraic geometry, Mirror Symmetry, Hodge theory, Degeneration.The author was supported in part by DFG SPP 1154 and Studienstiftung des dt. Volkes.


1



113

2 HELGE RUDDAT

Here Xt is a general fibre of X → S, Λ is the sheaf of integral tangent vectors on thecomplement B \Δ of the singular locus Δ of the affine structure and i : B \Δ → Bis the inclusion.

Starting from dimension four this result can not always hold for one expectsstringy Hodge numbers to replace ordinary Hodge numbers [1], [4]. In fact, theauthors of [20] impose the subtle condition that certain lattice polytopes encodingthe local affine monodromy are standard simplices rather than elementary simplices.By definition, a lattice simplex is elementary if it does not contain any interiorintegral points.

In general, the stringy Hodge numbers hp,qst are greater than or equal to the or-

dinary Hodge numbers. For a not necessarily maximal degeneration, we supplementthis to

hp,qaff (B) ≤ hp,q(Xt) ≤ hp,q

st (Xt).

Moreover, we observe that mirror symmetry interchanges differences: Let X →S be a maximal degeneration with n = dimXt = 4. We can recover the difference ofthe stringy to the ordinary Hodge numbers on the mirror dual degeneration X → Sas the difference of the ordinary to the affine Hodge numbers, i.e.,

hp,q(Xt)− hp,qaff (B) = hn−p,q

st (Xt)− hn−p,q(Xt).

Note that a zero difference on the left hand side under the standard simplexcondition of [20] is reflected by smoothness on the right hand side because each lat-tice polytope locally describes a toric singularity of Xt and smoothness correspondsto a standard simplex. Note that mirror symmetry of stringy Hodge numbers forcomplete intersections in toric varieties was shown in [5].

More generally, we investigate the Hodge groups for non-maximal degenera-tions by defining the new degeneration space classes hypersurface type (h.t.) andcomplete intersection type (c.i.t.). For instance, an anticanonically embedded gen-eral hypersurface in a Fano toric variety yields a h.t. degeneration. To refine this toa maximal degeneration one would have to form its MPCP resolution and possiblyeven blow this further up.

We relate the hp,q(Xt) to the logarithmic Hodge numbers of the central fibrehp,qlog(X0) and derive a recipe to compute hp,q

log(X0) in terms of hp,qaff (B) and additional

contributions which we call log twisted sectors. The latter depend on the affine dataof B (the monodromy polytopes) as well as a continuous parameter Z which is thelocus of the logarithmic singularities of X0. Our result was inspired by [6], whereBorisov and Mavlyutov give a conjectural definition of string cohomology for ahypersurface Calabi-Yau in a toric variety. They use toric Jacobian rings whichcome up in our setting as well. More recently, Helm and Katz [22] have relatedthe cohomology of a subvariety of a torus to the topology of the tropical varietyobtained from a normal crossing degeneration thereof.

If f is a local equation for an irreducible component Zω of Z, Δω the corre-sponding monodromy polytope and C(Δω) the cone over the polytope then thegraded dimensions of the toric Jacobian rings R0(C(Δω), f) and R1(C(Δω), f) inthe notation of [6] play a central role in the computation of the log twisted sectors.For the moment, let X = Xω denote the smallest stratum of X0 containing Zω.For the canonical linear system V ⊆ Γ(X,OX(Zω)) given by f and its logarithmicderivatives, we set R(Zω)n := coker

(V ⊗Γ(X,OX((n−1)Zω)) → Γ(X,OX(nZω))

).

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LOG HODGE GROUPS ON A TORIC CALABI-YAU DEGENERATION 3

If Δω is a simplex, we have V = Γ(X,OX(Zω)), R(Zω) ∼= R0(C(Δω), f) and

dimR(Zω)n = #

{lattice points of n · Δω which are not a sum of a

lattice point of (n− 1) · Δω and a vertex of Δω

}

We prove the following result for the logarithmic Hodge groups Hq(X0,Ωplog)

of the central fibre X0 of a toric degeneration.

Theorem 0.1. Let X0 be a hypersurface type (h.t.) toric log Calabi-Yau space.

a) For each p, there is a spectral sequence which computes the logarithmicHodge groups Hq(X0,Ω

plog) whose E1 term can be given explicitly in terms

of i∗∧r Λ⊗Z k,

∧r i∗Λ⊗Z k and R(Zω ∩Xτ ) for various ω, τ .b) If every Δω is a simplex, the spectral sequence is degenerate at E2 and

dimEq,02 (Ωp

log) = hp,qaff (B)∑

k>0 dimEq−k,k2 (Ωp

log) = log twisted sectors

To relate this to the ordinary Hodge groups of the general fibre, the authors of[20] have shown for maximal degenerations that

(0.1) Hq(X0,Ωplog)

∼= Hq(Xt,Ωplog) = Hq(Xt,Ω

pXt

)

by means of a base change result for Hk(X ,Ω•log). The base change easily general-

izes to c.i.t. degenerations, so that generalizing (0.1) is equivalent to showing thatthe first hypercohomology spectral sequence computing Hk(X0,Ω

•log) degenerates

at E1. The classical way would be to show that Ω•log carries the structure of a

cohomological mixed Hodge complex ([13], 8.1.9). This requires the topologicalresult that H

k(X0,Ω•log) computes the cohomology of the Kato-Nakayama space

corresponding to X0 which we leave for future work. Instead, we show the degen-eration directly under some conditions up to dimX = 4.

The structure of this paper is as follows. All central results can be found in Section1 where we quickly recall Gross and Siebert’s constructions, give the main defini-tions h.t. and c.i.t. (1.1), state the result about the spectral sequence computingthe log Hodge groups (1.2) and give the base change result for the affine Hodgenumbers and its consequences for the base change of the ordinary Hodge numbersin low dimensions and a mirror result on the stringy Hodge numbers (1.3). InSection 2, we first derive some further consequences of the c.i.t. definition (2.1),in particular a set of inner monodromy polytopes which we then use to general-ize Gross-Siebert’s construction of local models (2.2), the exactness of C •(Ωp) andsome further technical properties. In Section 3, we treat Koszul cohomology fora semi-ample divisor Z in a toric variety and compute its cohomlogy in terms ofR(Z). In 3.3, we compare R(Z) with Jacobian rings in the case where the corre-sponding polytope is a simplex. In particular, we give a monomial basis for theJacobian ring of a non-degenerate Z. In 3.4, we identify the intermediate cokernelsof the Koszul complex with differential forms having poles on the toric boundaryand zeros along Z. These coincide with the summands of C •(Ωp), such that theKoszul complex leads to a resolution of these as we show in 4.1. In 4.2, we treatthe dependency of the choice of a vertex for the Newton polytope in the resolution.The central result about the computation of the log Hodge groups is then provedin 4.3-4.5. The basechange and mirror symmetry for the twisted sectors is thecontents of Section 5.

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4 HELGE RUDDAT

I would like to thank my PhD advisor Bernd Siebert for many useful discussionsand his support during my thesis. I also thank Mark Gross for supporting mycoming to the UCSD and helpful input and corrections. I am grateful to KlausAltmann, Renzo Cavalieri and Klaus Hulek for invitations to give a talk on mywork. I had inspiring meetings with Stefan Muller-Stach and Stefan Waldmann. Iowe to the Mathematische Fakultat der Albert-Ludwigs-Universitat for a providinga working environment.

1. Definitions and central results

1.1. Toric log Calabi-Yau spaces of hypersurface and complete inter-section type. We fix an algebraically closed field k. Recall from ([19], Def. 4.1)that a toric degeneration is flat family X → S = SpecA for some discrete valuationring A with residue field k such that

a) the generic fibre Xη is a normal algebraic space,b) the special fibre X0 is a union of toric varieties glued along toric boundary

strata andc) there is a closed subset Z ⊂ X of relative codimension at least two,

such that every point in X\Z has a neighbourhood which is etale locallyequivalent to an affine toric variety where X0 is identified with the toricboundary divisor and the deformation parameter is given by a monomial.

The Cartier divisor X0 in X induces a divisorial log structure on X which onemay pull back to X0 to turn it into a log space. For log structures, see [25], [19].The definition of a toric log Calabi Yau space ([19], Def. 4.3) (short: toric log CYspace) is precisely made such that X0 with its log structure is the key example. Theauthors of [19] demonstrate how to derive the dual intersection complex (B,P)from X0 which is a real affine manifold B with singularities in codimension twoand a polyhedral decomposition P with some further properties. Given lifted openglueing data s for (B,P), one can reconstruct X0 from the triple (B,P, s). Onemight even start directly with such a triple to construct a toric log CY spaceX0(B,P, s) (if one also adds a suitable log structure). Recall that a toric logCY space is positive if the section of the log smooth structure moduli bundle onX0\Z extends to X0 by attaining zeros rather than poles ([19], Def. 4.19). Ananalogous notion of positivity for (B,P) is a condition on the local monodromyaround the singular locus ([19], Def. 1.54). Recall that the set of polytopes P canbe considered as a category consisting of lattice polytopes as objects and inclusionsof faces as morphisms. We recall additional notation:

• P [l] for the subset of cells of dimension l,• Δ for the discriminant locus of B,• Λ, Λ for the local system of integral tangent and cotangent vectors on

B\Δ,• i : B\Δ → B for the natural inclusion,• Λτ for the subset of tangent vectors parallel to τ ∈ P at a relative interiorpoint of τ and

• Λτ for Λ⊥τ .

Recall that a pair (ω, ρ) ∈ P [1] × P [dimB−1] determines a loop around thesingular locus by going from one vertex of ω through the interior of one neighbouringmaximal cell of ρ to the other vertex of ω and returning to the first vertex by

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passing through the interior of the other maximal neighbouring cell of ρ. The orderof vertices and maximal cells and thus the orientation of the loop can be chosen byfixing integral primitive vectors dω ∈ Λω and dρ ∈ Λρ. It was shown in loc.cit. thatthe monodromy in a nearby stalk of Λ along the so determined homotopy class ofloop has a special shape and can be given by n → n + κωρ〈n, dρ〉dω where κωρ is

an integer independent of the choices of dω and dρ ([21], before Def. 1.4).Recall that a special fibre X0 of a toric degeneration is always positive. We will

assume from now on that X is a positive toric log CY space. The dual intersectioncomplex (B,P) is then also positive, i.e., each κωρ ≥ 0.

Recall that the inner monodromy polytope for ρ ∈ P [dimB−1] is constructedby fixing a vertex v ∈ ρ and by taking the convex hull of all mρ

v,v′ where v′ is a

vertex of ρ and n → n + 〈n, dρ〉mρv,v′ is the monodromy transformation of a stalk

of Λ near v for a loop going from v to v′ through the interior of the maximal cellon which dρ is negative and returning through the other one. It is denoted by

Δρ ⊂ Λρ ⊗Z R.

By restricting to vertices in a face τ of ρ, one gets for each e : τ → ρ a polytope

Δρ,e ⊂ Λτ⊗ZR which is a face of the previous one. It is clear that themρv,v′ are sums

of appropriate (κωρdω)’s. Up to integral translation, the monodromy polytopes areindependent of v. Dually, we have the outer monodromy polytopes

˜Δω ⊂ Λω ⊗Z R and ˜Δω,e ⊂ Λτ ⊗Z R

given ω ∈ P [1], resp. e : ω → τ . These are constructed from the monodromyof a stalk of Λ in some maximal cell σ containing ω along loops passing throughthe vertices of ω into other maximal cells σ′. The transformations have the shapem → m+ 〈dω,m〉nσ,σ′

ω . We have decorated the polytopes by ˜ in contrast to [19]to distinguish them from similar polytopes coming up later on.

Recall that there is a contravariant correspondence of closed strata Xτ of Xand cells τ ∈ P. The irreducible components of X are Xv for v ∈ P [0]. Becauseeach stratum is a toric variety, we also get a decomposition of X in a disjoint unionof locally closed strata

X =∐τ∈P

Int (Xτ )

where Int (Xτ ) is supposed to be the open torus in Xτ . For each ω ∈ P [1] there is

a possibly empty or non-reduced Cartier divisor Zω in Xω such that

Z =⋃

ω∈P[1]

Zω

is the log singular locus of X. We have Zω = ∅ if and only if ω doesn’t meet Δ. Fora semi-ample Cartier divisor (i.e. one whose invertible sheaf is generated by globalsections) E on a toric variety we denote its Newton polytope defined via a linearlyequivalent toric divisor by Newton (E). For subvarieties E of codimension greaterthan one, we set Newton (E) = {0}. Recall from [19] that we have

Newton (Zω) =˜Δω.

We write Zredω for the reduction of the effective Cartier Divisor Zω and set Zω :=

Zredω . We follow [2], [10] and call a semi-ample divisor E on a toric variety non-

degenerate if Newton (E) up to translation coincides with the convex hull of all

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6 HELGE RUDDAT

monomials with nontrivial coefficients given an equation of E in a toric chart andE has a regular or empty intersection with every torus orbit.

Definition 1.1. A positive toric log CY space is of hypersurface type (short:h.t.) iff

(1) The divisor Zω is non-degenerate for each ω ∈ P [1] and for some aω ∈ N≥1

Zω = aω · Zω.

(2) For each τ ∈ P, the set {Zω ∩Xτ |ω ∈ P [1], Zω ∩ Int (Xτ ) �= ∅} is eitherempty or contains only one element which we then denote by Zτ .

The nomenclature is deduced from Batyrev’s mirror construction [3]. A toricdegeneration which is fibrewise embedded as an anticanonical hypersurfaces in aFano toric variety in generic position yields an example of a h.t. space. Generally,having an embedding is not necessary of course. We will mostly concentrate onthe h.t. property in this paper. For the more general parts, we use the analogueof the Batyrev-Borisov mirror construction [7] as in the upcoming definition. Wecall a set of lattice polytopes Δ1, ...,Δr in an R-vector space W transverse if theirtangent spaces form an interior direct sum in W .

Definition 1.2. A positive toric log CY space is of complete intersection type(short: c.i.t.) iff

(1) The divisor Zω is non-degenerate for each ω ∈ P [1] and for some aω ∈ N≥1

Zω = aω · Zω

(2) For each τ and ω1, ω2 ∈ P [1], we have

{0} �= Newton (Zω1∩Xτ ) = Newton (Zω2

∩Xτ ) ⇒ Zω1∩Xτ = Zω2

∩Xτ

(3) For each τ ∈ P, the set {Newton (Zω ∩Xτ ) |ω ∈ P [1], Zω ∩ Int (Xτ ) �= ∅}is either empty or contains at most min(dim τ, codim τ ) many elementsΔτ,1,...,Δτ,q which are transverse. The corresponding divisors are denotedby Zτ,1, ..., Zτ,q.

1.2. A spectral sequence to compute the log Hodge groups. In this andin the next section, we are going to summarize the main results of the paper. Werecall some notions of [20] in the following, in particular the barycentric resolutionof the log Hodge sheaves. Let X a toric log CY space and j : X\Z → X denotethe canonical inclusion of the log smooth locus.

Definition 1.3. The log Hodge sheaf Ωr of degree r is the pushforward of thesheaf of log differential forms, i.e.,

Ωr := j∗Ωr(X\Z)†/k† .

The log Hodge group of index p, q is the cohomology group

Hp,qlog (X) := Hq(X,Ωp).

The log Hodge number of index p, q is hp,qlog(X) := dimHp,q

log (X).

Where useful, we will write ΩpX for Ωp. Recall from [20] that Fs(τ0 → τk) :

Xτk → Xτ0 is the inclusion of one stratum of X in another indexed by τ0, τk ∈ P.It is written this way to account for the possibly non-trivial glueing data s. We

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drop the base scheme S in the usual notation FS,s because we always assumeS = Speck. Recall that Fs(e) = F (e)◦se where F (e) is the standard toric inclusionand se ∈ Aut (Xτ2) is given by the action of a torus element. We set

Ωrτ := (κτ )∗κ

∗τ (q

∗τΩ

r/Tors)where qτ : Xτ → X and κτ : Xτ\(Dτ ∩ q−1

τ (Z)) → Xτ are natural inclusions withDτ = Xτ\Int (Xτ ).

Definition 1.4. We recall the barycentric complex given by

C k(Ωr) =⊕

τ0→...→τk

(qτk)∗((Fs(τ0 → τk)∗Ωr

τ0)/Tors)

where Tors is the torsion submodule. The differential is

(dbct(α))τ0→...→τk+1= ατ1→...→τk+1

+∑k

i=1(−1)iατ0→...τi→...→τk+1

+(−1)k+1Fs(τk → τk+1)∗ατ0→...→τk .

The proof of the following lemma will be given in Section 2.3.

Lemma 1.5. If X is c.i.t., there is an exact sequence

0 → Ωr → C 0(Ωr)dbct−→ C 1(Ωr) → ... .

Each morphism e : τ1 → τ2 in P can be identified with an edge in B and wedefine the open set We ⊂ B as the union of all relative interiors of simplices in thebarycentric subdivision of P which contain e. We set Ze = Zτ1 ∩ Xτ2 and thenhave R(Ze) as before defined with respect to the toric variety Xτ2 . We prove parta) of the following result in Section 4.3 and part b) in Section 4.5.

Theorem 1.6. Let X be a h.t. toric log CY space. We fix r.

a) The E1 term of the hypercohomology spectral sequence of C •(Ωr) is

Ep,q1 : Hq(X,C p(Ωr)) ⇒ H

p+q(X,C •(Ωr)) = Hp+q(X,Ωr),

where

Hq(X,C p(Ωr)) =⊕

e:τ0→...→τp

⎧⎪⎪⎨⎪⎪⎩

Γ(We, i∗∧r Λ⊗Z k) for q = 0,

R(Ze)q ⊗Γ(We, i∗

∧r+q Λ⊗ k)

Γ(We,∧r+q

i∗Λ⊗ k)for q > 0

⎫⎪⎪⎬⎪⎪⎭

.

Note that Lemma 4.18 gives the differential d1.b) If every Δω is a simplex, the spectral sequence in a) is degenerate at E2

level and

Ep,02 = Hp(B, i∗

r∧Λ⊗ k).

1.3. Base change of the logarithmic Hodge groups. In Section 2.2 wegive for each point x of a c.i.t. space X a local model for the log structure, i.e., anaffine toric variety Yloc with a toric Cartier divisor Xloc, s.t. at x, X is etale locallyequivalent to an open subset of Xloc, and the log structure on X agrees with thepullback to Xloc of the divisorial log structure on Yloc given by the divisor Xloc.This is important for points in Z, the others fulfil this by definition.

Analogous to ([20], Def. 2.7), we say that a toric deformation X → S whereX = X0 is a c.i.t. space is a divisorial deformation of X if it is etale locallyisomorphic to the c.i.t. local models Yloc. We are then going to prove:

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8 HELGE RUDDAT

Theorem 1.7. Let π : X → SpecA be a divisorial deformation of a c.i.t.toric log CY space, j : X\Z → X the inclusion of the log smooth locus and writeΩ•

X := j∗Ω•X †/A† . Then for each p, Hp(X ,Ω•

X ) is a free A-module, and it commutes

with base change.

Corollary 1.8. Let π : X → SpecA be a divisorial deformation of a c.i.t.toric log CY space X. If the log Hodge to log de Rham spectral sequence on X (i.e.,the hypercohomology spectral sequence of Ω•

X) degenerates at E1 then Hq(X ,ΩpX )

is a free A-module, and it commutes with base change.

Proof. By Grothendieck’s cohomology and base change theorem, it sufficesto prove surjectivity for the restrictions Hq(X ,Ωp

X ) → Hq(X,ΩpX). This means

surjectivity for E1(Ω•X ) → E1(Ω

•X). Since degeneration is an open property, both

spectral sequences are degenerate at E1 and we are done if we show surjectivityof (GrFH

k(X ,Ω•X ))/Tors → GrFH

k(X,Ω•X) where F is the Hodge filtration. This

follows from Thm. 1.7 by the surjectivity of Hk(X ,Ω•X ) → Hk(X,Ω•

X). �Remark 1.9. If all inner monodromy polytopes are simplices then the generic

fibre Xη is an orbifold. The restriction of ΩrX ⊗OSpec A

OSpec η coincides with thepushforward of Ωr

(Xη\SingXη)/kto Xη. By [33], these sheaves give the natural mixed

Hodge structure on Xη ([13]) which is pure in each cohomology degree.

Definition 1.10. The affine Hodge group of degree (p, q) of a toric log CYspace X, resp. its dual intersection complex (B,P), is defined as

Hp,qaff (X) = Hp,q

aff (B) = Hq(B, i∗

p∧Λ⊗ k).

We denote its dimension by hp,qaff (X) and call it affine Hodge number.

We are going to prove the following result in Section 5.1.

Theorem 1.11. Let X be a c.i.t. toric log CY space.

a) For each p, q there is a natural injection

Hp,qaff (X) ↪→ Hp,q

log (X).

b) For each k there is a natural injection⊕p+q=k

Hp,qaff (X) ↪→ H

k(X,Ω•)

which is compatible with the canonical filtration induced on Hk(X,Ω•).

Corollary 1.12. Let Xt be a general fibre of a toric degeneration with at mostorbifold singularities. Assume that the central fibre X is a c.i.t. space. For all p, q,we have

hp,qaff (X) ≤ hp,q(Xt).

Proof. By Thm. 1.7, Thm. 1.11 and Remark 1.9, we have

hp,qaff (X) ≤ dimGrpFH

p+q(X,Ω•X) ≤ rkGrpFH

p+q(X ,Ω•X )/Tors = hp,q(Xt)

where F is the Hodge filtration on Ω•X . �

In Section 5.2, we give a proof of the following result.

Theorem 1.13. Let X be a h.t. toric log CY space. Assume we have one ofthe following conditions

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a) dimX ≤ 2b) dimX = 3, each Δτ is a simplex and every component of Δ\Δ0 is con-

tractible where Δ0 denotes the set of points in Δ where the correspondingmonodromy polytope Δτ has dimension two

c) dimX ≤ 4 and each Δτ is an elementary simplex

Then the log Hodge to log de Rham spectral sequence degenerates at

Ep,q1 : Hp(X,Ωq) ⇒ H

p+q(X,Ω•).

Remark 1.14. To prove the degeneration of the log Hodge to log de Rhamspectral sequence in greater generality, a common way would be show that Ω•

X

carries the structure of a cohomological mixed Hodge complex ([13], 8.1.9). Inparticular, this requires a Z-structure which one would obtain as the pushforwardfrom the semi-analytic Kato-Nakayama space X → X. One then needs to showthat Ω•

X is quasi-isomorphic to a pushforward of a modified de Rham complex on

X which is in turn a resolution of Z⊗ZC on X. We leave the topological propertiesof the local models to future work.

Theorem 1.15. Assume that we are given a h.t. space X and that Xt isa general fibre of a degeneration into X. Assume we are in one of the cases ofThm. 1.13 and that Xt is an orbifold, i.e., each Δτ is a simplex. We have for eachp, q,

a) hp,qlog(X) = hp,q(Xt)

b) If we are in case a) or c), we have

hp,q(Xt)− hp,qaff (X) = hn−p,q

st (Xt)− hn−p,q(Xt).

Example 1.16. Note that Theorem 1.15, a) holds for all Calabi-Yau threefoldsobtained from simplicial subdivisions of reflexive 4-polytopes where the subdivisiondoesn’t introduce new vertices. In particular, we obtain for the quintic threefold Xin P3 as well as for its mirror dual orbifold the affine Hodge diamond

10 0

0 1 01 1 1 1.

The log twisted sectors of X contribute to h2,1(X) = h1,2(X) = 101 by adding 100to the affine Hodge numbers and since X is smooth, hp,q(X) = hp,q

st (X). All logtwisted sectors of X are trivial. On the other hand, we obtain non-trivial orbifoldtwisted sectors in degree (1, 1) and (2, 2). We have h1,1

st (X) = h1,1(X) + 100 =

h1,1aff (X) + 100 = 101 and the analogous for h2,2

st (X).

2. Local models for c.i.t. spaces

2.1. Reduced inner monodromy polytopes. The c.i.t. property is a gen-eralization of h.t. becoming distinct only if dimX ≥ 4. It also generalizes simplicity([19], Def. 1.60, Rem. 1.61) which we referred to in the introduction as a maximaldegeneration. There is a natural bijection

P ↔ { vertices of the barycentric subdivision of P}by identifying a cell with its barycenter. Moreover, there is a natural bijectionbetween the set of d-dimensional simplices in the barycentric subdivision and the

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10 HELGE RUDDAT

set of chains of proper inclusions τ0 → ... → τd of cells in P. It follows from([19], Def. 1.58) that the discriminant locus Δ is the union of all codimension twosimplices in the barycentric subdivision of P corresponding to chains of the shapeω → ... → ρ with ω ∈ P [1], ρ ∈ P [dimB−1] and κωρ �= 0.

Lemma 2.1. Let X be a c.i.t. toric log CY space. Fix ω ∈ P [1]. The barycentricedge corresponding to some e : ω → τ is contained in Δ if and only if

Zω ∩ Int (Xτ ) �= ∅.

Proof. We just sketch the proof to keep the notation concise. The Newtonpolytope of the closure of Zω ∩ Int (Xτ ) is a face of Δω contained in a translate ofτ⊥. The intersection is non-trivial if and only if this Newton polytope has positivedimension. This happens if and only if it contains an edge of Δω which in turncorresponds to some τ → ρ such that this edge is parallel to ρ⊥. This meansκωρ �= 0. This happens for some e : τ → ρ if and only if e is contained in Δ. �

Lemma 2.2. Let X be a c.i.t. toric log CY space.

a) We haveκω1ρ

aω1

=κω2ρ

aω2

,

whenever the barycentric edges ω1 → ρ, ω2 → ρ are contained in Δ.b) We define aρ as the integral length of Newton ((Z ∩Xρ)

red ) and have for

each (ω, ρ) ∈ P [1] × P [dimB−1]

κωρ = aωaρ.

Proof. We prove a). Because codim ρ = 1 there is at most one Δρ,i by

Def. 1.2. Assume we have P [1] � ωjej−→ ρ for j = 1, 2 such that Zωj

∩ Xρ �= ∅.By Lemma 2.1, this is equivalent to e1, e2 ∈ Δ. We get Newton (Zωj

∩Xρ) = Δρ.

Becauseκωj,ρ

aωjis the integral length of Δρ,1, we get the assertion. Part b) is just

rephrasing this. �

Remark 2.3. (1) A positive toric log CY space in dimension 2 where Zis reduced is h.t.. Not included in the h.t. definition are situations wheresome Zω is the union of a reduced point and a double point, for instance.Two double points, however, would fulfill h.t. by having aω = 2.

(2) If X is simple then X is h.t. iff for each τ ∈ P the number of outer (orinner) monodromy polytopes at τ given in the simplicity definition is lessor equal than one. The inverse direction follows from the multiplicative

condition for the log structure∏

ω dω ⊗ fετ (ω)ω |Vτ

= 1 ([19], Thm 3.22)which implies that all Zω|Xτ

for varying ω are either empty or agreebecause fω is a local equation of Zω. In particular, if X is simple ofdimension 3 then X is h.t..

(3) Why do we allow aω > 1? This is best seen by the just mentioned multi-plicative condition for the log structure. If some inner simplex polytopehas non-primitive edges of different integral lengths, we have to requiresome aω > 1 for a log structure to exist on such a space.

(4) Recall that a discrete Legendre transform interchanges inner and outermonodromy polytopes. It also interchanges aω and aρ and we will see thatthere is a collection of reduced inner monodromy polytopes analogous to

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the collection of reduced outer monodromy polytopes in the definition ofc.i.t.

Here is a lemma which relates the inner and outer monodromy polytopes to

the κωρ. It is directly deduced from the construction of Δρ and ˜Δω.

Lemma 2.4. (1) Given ρ ∈ P [dimB−1], there is a natural surjection

{ω → ρ |ω ∈ P [1], κωρ �= 0} → {edges of Δρ}.Moreover, ω is collinear to the edge it maps to and κωρ is its integrallength.

(2) Given ω ∈ P [1], there is a natural surjection

{ω → ρ | ρ ∈ P [dimB−1], κωρ �= 0} → {edges of ˜Δω}.Moreover, a translate of ρ⊥ contains the edge it maps to and κωρ is itsintegral length.

Lemma 2.5. For N = Zn and M = Hom(N,Z), let Σ be a complete fan inNR = N ⊗Z R and ψ a piecewise linear function on NR with respect to Σ. Assumethat ψ comes from a lattice polytope Ξ ⊂ MR, i.e.,

ψ(n) = −min{〈m,n〉 |m ∈ Ξ}.Given ω ∈ Σ[n−1], let σ1, σ2 ∈ Σ[n] the two maximal cones containing ω. We setκω = integral length of m1−m2 where m1,m2 ∈ M with m1 = ψ|σ1

and m2 = ψ|σ2.

The data

k : Σ[n−1] → N, k(ω) = κω

determines Ξ uniquely up to translation.

Proof. Note that m1−m2 is collinear to ω⊥ and is thus uniquely determinedby κω up to orientation. The combinatorics of the fan now gives a recipe to assemblethese edge vectors to the polytope Ξ. Fix some maximal cone v0 ∈ Σ[n]. To eachchain γ of the shape

v0 ⊃ ω0 ⊂ v1 ⊃ ω1 ⊂ ... ⊃ ωl ⊂ vl

with ωi ∈ Σ[n−1], vi ∈ Σ[n], set mγ =∑l

i=0 mi where mi is the unique element inM which is collinear to ω⊥

i , has integral length κωiand evaluates positive on the

interior of vi. We obtain

Ξ = convex hull of {mγ | γ is a chain}.�

The following proposition takes care of the inner monodromy polytopes whichare not obvious from the definition of c.i.t. unlike the outer ones.

Proposition 2.6. Let X be c.i.t. and τ ∈ P. Let Δτ,1, ..., Δτ,q be the associ-ated set of Newton polytopes. There exists a canonical set of lattice polytopes

Δτ,1, ...,Δτ,q ⊂ Λτ ⊗Z R

such that, for each ρ ∈ P [dimB−1], e : τ → ρ and Δρ,e non-trivial, we find a unique

i such that Δρ,e is an integral multiple of Δτ,i.

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12 HELGE RUDDAT

Proof. The correspondence: We have fixed τ . All ω’s are supposed to be in

P [1] and all ρ’s in P [dimB−1]. Consider the diagram

{ω → τ → ρ |κωρ �= 0} ��

��

{τ → ρ | Δρ,τ→ρ �= 0}

��

{ω → τ |Zω ∩ Int (Xτ ) �= ∅} �� {1, ..., q}.

The upper horizontal arrow is just “forgetting ω”, the left vertical one is “forgettingρ” and uses Lemma 2.1. The lower horizontal map is given by part (3) of thedefinition of c.i.t.. There is only one way to define the dotted arrow to make thediagram commute and we need to argue why it is well-defined. Assume we haveω1 → τ → ρ and ω2 → τ → ρ with κω1ρ �= 0 �= κω2ρ. By Lemma 2.4 we find that˜Δω1

and ˜Δω2both have an edge contained in a translate of the straight line ρ⊥.

The same holds for ˜Δω1,ω1→τ ,˜Δω2,ω2→τ and also for 1

aω1

˜Δω1,ω1→τ ,1

aω2

˜Δω2,ω2→τ

which are the Newton polytopes of Zω1∩ Xτ and Zω2

∩ Xτ , respectively. Thus,these polytopes cannot be transverse and by (3) of the c.i.t. definition they haveto be the same up to translation. This makes the dotted map well-defined.

We denote the preimage of i under the lower horizontal map by Ωτ,i.Defining the Δτ,i: We stay with the previous setup. We define

Δτ,i :=1

aρΔρ,τ→ρ

where i is the image of Δρ,τ→ρ under the dotted arrow. It is easy to see that (up totranslation) this is a lattice polytope where an edge which is the image of some ω viaLemma 2.4 has length aω. We have to show that we get the same Δτ,i if we chooseanother τ → ρ′ with κωρ′ �= 0 to define it. We are going to apply Lemma 2.5. By

([19], Remark 1.59), both 1aρΔρ,τ→ρ and 1

aρ′Δρ′,τ→ρ′ give piecewise linear functions

on Στ , the normal fan of τ in Hom (Λτ ,Z) ⊗Z R. We have an inclusion reversingbijection

cones in Στ ↔ faces of τ.

Codimension one cones ω ∈ Σ[dim τ−1]τ correspond to edges ω of τ . So this is

consistent with the notation in the lemma. Note that the data k : Σ[dim τ−1]τ → N

is the same for both polytopes because for each ω ∈ Σ[dim τ−1] we have

κω =

{aω if ω → τ ∈ Ωτ,i

0 otherwise.

We deduce that 1aρ′

Δρ,τ→ρ and frac1aρ′Δρ′,τ→ρ′ coincide up to translation. �

We extract a definition from the previous proof.

Definition 2.7. Given a c.i.t. X and τ ∈ P. For 1 ≤ i ≤ q, we define

Ωτ,i = {ω → τ | ˜Δω,ω→τ = a · Δτ,i for some a > 0}

Rτ,i = {τ → ρ | Δρ,τ→ρ = a ·Δτ,i for some a > 0}where Δτ,i is the polytope given in Prop. 2.5.

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Note that for ω → τ, τ → ρ we have κωρ �= 0 if and only if there is some i suchthat ω → τ ∈ Ωτ,i and τ → ρ ∈ Rτ,i which can be deduced from the diagram in theproof of Prop. 2.6. In view of ([19], Def. 1.60), we see that this property generalizesfrom simplicity to c.i.t. spaces.

2.2. Toric local models for the log structure. In this section, we give adirect generalization of the local model construction developed by M. Gross andB. Siebert in [20] to the c.i.t. case. The proof will remain sketchy where thereis little difference to loc.cit.. Recall Construction 2.1 in loc.cit. where Y is theproduct of a torus with the affine toric variety given by the cone over the Cayleyproduct of τ and the Δτ,i and X is the invariant divisor given by the rays in τ . Weprefer to call the spaces X,Y of loc.cit. Xloc, Yloc at this point.

Proposition 2.8. Suppose we are given a c.i.t. toric log CY space X and ageometric point in the log singular locus x → Z ⊆ X, there exist data τ, ψ1, ..., ψq

as in ([20],Constr. 2.1) defining a monoid P and an element ρ ∈ P , hence affine

toric log spaces Y †loc, X

†loc → Speck†, such that there is a diagram over Speck†

V †

��

�� φ

��

��

�

X† X†loc

with both maps strict etale and V † an etale neighbourhood of x.

Proof. As in loc.cit., we take the unique τ ∈ P such that x ∈ Int (Xτ ). By thedefinition of c.i.t., we then have the outer monodromy polytopes Δτ,1, ..., Δτ,q ⊂ Λτ .By Prop. 2.6, we also obtain Δτ,1, ...,Δτ,q ⊂ Λτ . By renumbering, we may assumethat x ∈ Zτ,1, ..., Zτ,r and x �∈ Zτ,i for r < i ≤ q. We set

Δi =

{Δτ,i for 1 ≤ i ≤ r{0} for r < i ≤ dimB − dim τ

We redefine q = dimB−dim τ . The polytopes Δi give piecewise linear functions ψi

on the normal fan Στ in N ′R= Hom(Λτ ,Z)⊗ZR. By ([20], Constr. 2.1), we obtain a

monoid P ′ ⊆ N ′ with P ′ = C(τ )∨∩N ′, a monoid P ⊆ N = N ′⊕Zq+1, ρ ∈ P givenby ρ = e∗0, Yloc = Spec k[P ] and Xloc = Spec k[P ]/(zρ). To obtain a log-structureon Xloc, we use the pullback of the divisorial log structure given by Xloc in Yloc.To proceed as in the proof of ([20], Thm. 2.6), we choose g : τ → σ ∈ P [dimB] tohave an etale neighbourhood V (σ) of x. We are going to construct a diagram withstrict etale arrows

V (σ)†

pσ

��

��

�V (τ )†� �� V †� ��

φ

��

��

X† X†loc.

Recall from ([19], Thm. 3.27) that pulling back the log-structure from X† to V (σ)gives a tuple

f = (fσ,e)e:ω→σ ∈ Γ

(V (σ),

⊕e:ω→σdim ω=1

OVe

)

125

14 HELGE RUDDAT

where Ve is the closure of Int (Xω) in V (σ). We write fω for fσ,e with e : ω → τ .Let x[σ] be the unique zero-dimensional torus orbit in V (σ). We may assume thatf is normalized, i.e., fω|x[σ] = 1 for each ω. This is possible because, if f is notnormalized, we may use a pullback by an automorphism of V (σ) as explained in([19], after Def. 4.23) to obtain a normalized section. Let pσ : V (σ) → X be anetale map whose pullback-log-structure is the normalized section. Note that

p−1σ Zω = {fω = 0} ⊆ Ve

Let f redω be such that (f red

ω )aω = fω. Fixing some 1 ≤ i ≤ r, we claim that thefunctions f red

ω for ω → τ ∈ Ωτ,i glue to a function fi on⋃

e∈Ωτ,iVe. This will follow

if we show that the corresponding Zω glue because then their defining functionsf redω can at most differ by a non-trivial constant which is 1 by the normalizationassumption. To show this, we consider a diagram

ω1

��

��

� e1

τ ′ �� τg

�� σ

ω2

�� e2

��

which algebraic geometrically means that we have a stratum

Xτ ′ ⊂ Xω1∩Xω2

on which we wish to show Zω1∩ Xτ ′ = Zω2

∩ Xτ ′ . By the c.i.t. property (2), itsuffices to show

Newton (Zω1∩Xτ ′) = Newton (Zω2

∩Xτ ′).

This follows from the c.i.t. property (3) because Newton (Zω1∩ Xτ ′) and

Newton (Zω2∩Xτ ′) cannot be transverse due to the fact that

Newton (Zω1∩Xτ ) = Newton (Zω2

∩Xτ ) = Δτ,i

is a non-trivial face of each. Thus, we have functions fi as claimed before. We cannaturally extend these to functions on V (τ ) which we are also going to denote byfi.

We now choose coordinates z1, ..., zq on Int (Xτ ) ∼= (k×)q, and pull these backto functions on V (τ ). By the c.i.t. property (3) we know that the Zτ,i meettransversely in v := p−1

σ (x), so we can find a subset {i1, ..., ir} ⊆ {1, ..., q} such thatF := det(∂fi/∂zij )1≤i,j≤r is invertible in v (see [14], Cor. 16.20). By reordering theindices, we can assume {i1, ..., ir} = {1, ..., r}. For r < i ≤ q, we set fi := zi − zi(v)so that all fi vanish in v and give a set of local coordinates at v when restricted toInt (Xτ ). We fix an isomorphism

V (τ ) ∼= Speck[∂P ′]× (k×)q.

We are going to choose V as a Zariski open neighbourhood of v in V (τ ). Before wesay which one exactly, we define the map φ : V → Xloc = Speck[∂P ′ ⊕ Nq] by

φ∗(zp) = zp for p ∈ ∂P ′

φ∗(ui) = fi for 1 ≤ i ≤ q

126


where ui is the monomial in k[∂P ′ ⊕ Nq] corresponding to the ith standard basisvector of Nq. Now V is chosen in a way such that φ is etale. That this can be doneis just a repetition of the argument in ([20], Prf. of Thm. 2.6). To see that thepullbacks of the two log-structures to V coincide, we check that both give the samesection in Γ(V,

⊕e:ω→σdim ω=1

OVe). Arguing as in ([20], Prf. of Thm. 2.6), this comes

down to showing, for each e : ω → τ and n ∈ N ′,

(dω ⊗ fω)(n) =

r∏i=1

f〈m−

i −m+i ,n〉

i

holds on the closed toric subset Ve where

• v−, v+ are the vertices of ω,• dω is the unique primitive vector pointing from v− to v+ and• m−

i ,m+i ∈ Hom(N ′,Z) are ψi|v− , ψi|v+ , respectively where v± is the max-

imal cone in Στ corresponding to v±.

We have that ψi bends at ω if and only if e ∈ Ωτ,i. By convexitym−i −m+

i is positiveon v− like dω. Combining this with the fact that the edge of Δi corresponding toω has length aω, this is just saying

m−i −m+

i =

{0 e �∈ Ωτ,i

aωdω e ∈ Ωτ,i.

By construction, we have that fi|Veis invertible for e �∈ Ωτ,i. Since we have chosen

f to be normalized and the invertible elements of a toric monoid ring k[P ] are givenas k× × P×, we have

fi|Ve=

{1 e �∈ Ωτ,i

f redω |Ve

e ∈ Ωτ,i.

Using (f redω )aω = fω, this finishes the proof. Note that we have slightly simplified

the proof as compared to loc.cit. by requiring f to be normalized in the beginning(this implied hp = 1 for all p in the notation of loc.cit.). �

The local models are the key ingredient to prove the base change for the hy-percohomology of the logarithmic de Rham complex:

Proof of Theorem 1.7. As argued in ([26], Lemma 4.1), one may assumethat A is a local Artinian k[t]-algebra. Then, using the existence of local modelsfrom Prop. 2.8, the proof becomes literally the same as in ([20], Thm. 4.1). �

2.3. The barycentric resolution of the log Hodge sheaves. The exis-tence of local models for c.i.t. spaces enables many of the further constructionsin [20]. The entire section 3.1 in ([20], Local calculations) doesn’t use simplicityarguments and extends directly to the c.i.t. case. In section 3.2, Prop. 3.8, Theo-rem 3.9, Cor. 3.10, Cor. 3.11, Lemma 3.12 and Lemma 3.13 are valid in the c.i.t.case. For this paper, we get three results from this. First, we obtain a proof ofLemma 1.5, i.e., have the exact sequence

0 → Ωr → C 0(Ωr) → C 1(Ωr) → ...

Roughly speaking, this is a resolution given by a kind of Deligne’s simplicial scheme:We pull back the sheaf to each toric stratum and then apply the inclusion-exclusion-principle to get a complex. Lemma 3.14 and Lemma 3.15 in loc.cit., however, failto be true in the c.i.t. setting. At least in the proof of 3.15 simplicity is being used

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16 HELGE RUDDAT

explicitly. In any event, we are not interested in these results. We just use a smallreplacement for Lemma 3.14:

Lemma 2.9. Given a c.i.t. space X, τ ∈ P, then v ∈ τ [0] induces a canonicalchoice of a vertex

Vert i(v) ∈ Δτ,i for each i.

Proof. Let v denote the maximal cone in the normal fan Στ corresponding

to v. Choose vi = Vert i(v) ∈ Δ[0]τ,i such that

vi −Δτ,i ⊆ v∨.

There is at most one such vi because otherwise v∨ would have to contain a straight

line. There exists one because ψτ,i, the piecewise linear function associated to Δτ,i,

is linear on v. In fact, we have ψτ,i|v = −〈vi, ·〉. �The third consequence for c.i.t. spaces which is most importance to us is ([20],

Prop. 3.17) which we cite here. Just note that dlog(fa) = a · dlogf has the samepoles as dlogf which is the reason why the sheaves of differentials only care aboutZred and not Z.

Proposition 2.10. Let X be c.i.t.. Given P [0] � vg→ τ1

e→ τ2, the image ofthe inclusion (Fs(e)

∗Ωrτ1)/Tors in Fs(e ◦ g)∗Ωr

v is

kern

(Fs(e ◦ g)∗Ωr

vδ−→

⊕i=1,...,qwi �=vi

Ωr−1(Z′

τ2,i)†/k†

)

where:

(1) We set vi = Vert i(v). The direct sum is over all i and all vertices wi ∈Δ

[0]τ1,i

with wi �= vi.

(2) Z ′τ2,i

= Fs(e)−1(Zτ1,i) which might be empty.

(3) We define a log structure on Xv as the pushforward of the pull-back viaXv\Z ↪→ X\Z (Z has codimension two in Xv). Then, Ωr−1

(Z′τ2,i)

†/k†is

defined by pulling back the log structure from X†v via Xτ1 and Zτ1,i to

Z ′τ2,i

.(4) For α ∈ FS,s(e ◦ g)∗Ωr

v, the component of δ(α) in the direct summand

Ωr−1(Z′

τ2,i)†/k†

corresponding to some wi is given by ι(∂wi−vi)α|(Z′τ2,i)

† .

In a setting more general than the simple case with standard outer monodromysimplices as dealt with in [20], the resolution C •(Ωr) is no longer acyclic. One of themain points of this paper is to produce an acyclic resolution for this complex. Westart off by constructing a resolution of a summand of C •(Ωr) on a single stratumfor which we are going to use a Koszul complex.

128


3. Koszul cohomology

This section is separate from the previous ones and we will be reusing some ofthe notations.

3.1. The general setting. Koszul cohomology is a globalized version of theKoszul complex for affine rings (see [14], Exc.17.19). It was extensively developedand exploited by Mark Green in [16] and [17].

Let X be a k-variety, L an invertible sheaf on X and V a finite dimensionallinear subspace of Γ(X,L) = Hom(OX ,L). Defining φ(1) as the canonical map(V ⊗OX → L) ∈ V ∗ ⊗ L, we get a complex

0 → OXφ→ L⊗

1∧V ∗ → L⊗2 ⊗

2∧V ∗ → L⊗3 ⊗

3∧V ∗ → ...

whose differential is given by α → φ(1) ∧ α. We introduce the notation

Ki(V,L) := L⊗i ⊗i∧V ∗

and call the according differential di. Form ∈ Z, we define the twists Ki(V,L,m) :=Lm ⊗Ki(V,L) and denote the dual by Ki(V,L,m).

Lemma 3.1. If V is base point free, i.e., V ⊗OX → L, s⊗g → s(g) is surjective,then for each m ∈ Z the Koszul complexes K•(V,L,m) and K•(V,L,m) are exact.

Proof. Since L is locally free, it suffices to show exactness for one m. Fur-thermore, it is enough to show it for K•(V,L), and exactness of the dual followsby applying Hom(·,OX) because the complex is locally free. Exactness can beconsidered at stalks. After choosing a basis {x1, ..., xn} of V , in the notation of[14], the Koszul complex is locally isomorphic to K(x1, ..., xn). Due to base pointfreeness of V , at each stalk at least one of the xi is invertible. By ([14], Prop.17.14a)) multiplication with xi annihilates the cohomology of K•(V,L) which is thustrivial. �

3.2. Semi-ample line bundles on toric varieties. In the following, we fixa toric variety X with character lattice M . A Cartier divisor Z on X is linearlyequivalent to a (non-unique) torus invariant Cartier divisor D. By a standard pro-cedure (see [15]) we associate to D its support function ψD which is a piecewiselinear function on the fan of X. By ([8], Prop. 6.7), semi-ampleness of Z is equiva-lent to the convexity of ψD. Convex piecewise linear functions, in turn, correspondto lattice polytopes Δ ⊂ M ⊗Z R whose normal fan can be refined to the fan of X,and Δ is uniquely associated to Z up to translation by lattice vectors. By ([15],Lemma in 3.4), one has

Γ(X,OX(D)) =⊕

m∈Δ∩M

k · zm.

An element of this corresponds to an effective divisor which is linearly equivalentto D. The correspondence is 1:1 up to the operation of k× on Γ(X,OX(D)). Theelement of the right hand side gives an equation of the divisor on the big torus as aLaurent polynomial. In the following, when we talk about the equation of a divisorwe mean a corresponding element f ∈

⊕m∈Δ∩M k · zm. Occasionally, we will

consider f as an element of the field of fractions Quot (OX) in which Γ(X,OX(D))

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18 HELGE RUDDAT

naturally embeds as {q | div(q) ≥ −D}. In the following, we always consider a fixedtranslation representative of Δ.

Lemma 3.2. Let Z be an effective divisor on X which is linearly equivalent toD and is given by an equation f . There are isomorphisms

⊕m∈Δ∩M

k ·zm· 1f−→ Γ(X,OX(Z)) and

⊕m∈Δ∩M

k ·(zm)∗·f−→ Γ(X,OX(Z))∗.

Proof. We have OX(D) = f · OX(Z) in the field of fractions of OX . Thisinduces the isomorphism of global sections and the one for their duals. �

Set N = Hom(M,Z), and choose some equation f =∑

m∈Δ∩M fmzm of adivisor Z. Using Lemma 3.2, we define the log derivation map

∂Z : (N ⊕ Z)⊗Z k → Γ(X,OX(Z))

via (n, a) → f−1∂(n,a)f = f−1∑

m∈Δ∩M fm〈(n, a), (m, 1)〉zm and denote its imageby V . Note that neither V nor ∂Z depends on the scalar multiple of an equation.However, ∂Z depends on the translation representative of Δ whereas V doesn’tdepend on it. In Remark 3.30, we discuss what happens if we move Δ. We definethe cone C(Δ) = {

∑v∈Δ[0] λv(v, 1) |λv ≥ 0} ⊂ (M ⊕Z)⊗Z R. where Δ[k] is the set

of k-dimensional faces of Δ. Let LC(Δ) denote the linear subspace generated by

C(Δ). We define TΔ = (LC(Δ) ∩ (M ⊕ Z)) ⊗Z k and think of it as the k-valuedtangent space of the cone.

Lemma 3.3. If Z is a semi-ample effective divisor on X with Newton polytopeΔ, then

0 → T⊥Δ → (N ⊕ Z)⊗Z k

∂Z−→ V → 0

is an exact sequence and thus dimV = dimΔ+ 1.

Proof. We show dimV ≥ dimΔ + 1, then the assertion follows becauseLC(Δ)⊥ ∩ ((M ⊕ Z) ⊗ k) clearly lies in the kernel. Let f be an equation of Z.By the hypotheses, fv �= 0 for v ∈ Δ[0]. Let {v1, ..., vn} be a dimΔ + 1 elementsubset of Δ[0] whose convex hull has dimension dimΔ. For each 1 ≤ i ≤ n, chooseni ∈ (N ⊕ Z) ⊗Z k such that 〈ni, (vj , 1)〉 = δij , the latter being the Kroneckersymbol. Consider the composition

(N ⊕ Z)⊗Z k → Γ(X,OX(Z)) →n⊕

i=1

kf−1zvi

where the last map is the natural projection⊕

m∈Δ∩M kf−1zm �⊕n

i=1 kf−1zvi .

The image of ni is fvif−1zvi . Therefore, the map is surjective, and we are done. �

We define the “tangent space” TΔ = {a · (x− y) | a ∈ k, x, y ∈ Δ[0]} and obtainan exact sequence

0 → TΔ → TΔh−→ k → 0

where h is the height function coming from the projection to the Z summand. Theabove lemma induces an isomorphism (∂Z)

∗ : V ∗ → TΔ. We may plug it into the

130


Koszul complex to get an isomorphism of complexes

. . . −−−−−→ Kl−1(V,O(Z),m)dl−1

−−−−−→ Kl(V,O(Z),m)dl−−−−−→ . . .

⏐⏐�id⊗

∧l−1(∂Z)∗⏐⏐�id⊗

∧l(∂Z)∗

. . . −−−−−→ OX((l − 1 +m)Z)⊗k

∧l−1 TΔ −−−−−→ OX((l +m)Z)⊗k

∧l TΔ −−−−−→ . . .

We denote the complex in the lower row by K•(TΔ, Z,m). This complex will playa major role in this paper. We now give an explicit description of its differential.

Lemma 3.4. The differential of K•(TΔ, Z,m) is

dZ : u⊗ α → u ·∑

m∈Δ∩M

fm1

fzm ⊗ (m, 1) ∧ α.

Note that in the lemma, we understand 1f z

m as a rational function.

Proof. Let us take a look at the composition Γ(X,OX(Z))∗ → V ∗ (∂Z)∗−→ TΔ.Its dual map sends (n, a) to

∑m∈Δ∩M fm〈(n, a), (m, 1)〉f−1zm, so we have

f(zm)∗ = (f−1zm)∗ → ((n, a) → fm〈(n, a), (m, 1)〉) = fm · (m, 1)

Now it is straightforward to see that the Koszul differential u⊗ α →u ·

∑m∈Δ∩M

1f z

m ⊗ ( 1f zm)∗ ∧ α transforms to dZ as given in the assertion. �

Recall the non-degeneracy definition from before Def. 1.1. The set of non-degenerate divisors form a Zariski open set in Γ(X,OX(D)) which can be deducedfrom Bertini’s theorem, see [8].

Lemma 3.5. If Z is non-degenerate, then V is base-point free.

Proof. See [2], Prop. 4.3. �

Lemma 3.6. Let Z be a semi-ample effective divisor on a toric variety X whoseNewton polytope is Δ. For m ∈ Z, we have

Hi(X,OX(mZ)) = 0 for 0 < i < dimΔ and for i > 0,m ≥ 0.

Proof. This is easy using the techniques of ([15], 3.5). The part for m ≥ 0is, in fact, the Corollary in loc.cit.. By similar arguments, the remaining part canbe reduced to showing that Hi(Rn,Rn\C; k) = Hi

C(Rn; k) = 0 for 0 < i < dimΔ

where C is either empty or a polyhedral cone whose largest linear subspace hasdimension codimΔ. The empty case is trivial, otherwise one may use Hi

C(Rn; k) =

Hi−1(Rn\C; k) for i > 1 and H0C(R

n; k) = H1C(R

n; k) = 0 via the long exactsequence of relative cohomology. There are two possibilities, either C is a linearsubspace or it is not. If it is not then its complement is contractible and we are done.If it is a linear subspace, its dimension is d = codimΔ. Then Hi−1(Rn\C; k) =Hi−1(Rn−d\{0}; k) which vanishes for (i− 1) < n− d− 1 ⇔ i < dimΔ. �

Proposition 3.7. Let Z be a non-degenerate divisor on a toric variety X withNewton polytope Δ. Set n = dimΔ+ 1 and HKi(V, Z,m) :=Hi

d•Γ(K•(V,OX(Z),m)). We have

HKi(V, Z,m) =

{0 for i �= nR(Z)n+m ⊗k

∧n V ∗ for i = n.

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20 HELGE RUDDAT

Remark 3.8. Using elementary results from Section 3.3 to show R(Z)n+m = 0for m ≥ 1, this generalizes the d = 1 case of the vanishing theorem [17], Thm. 2.2to toric varieties.

Proof. By Lemma 3.3, we have n = dimV ∗. The case Z = 0, i.e., R(Z)•>0 =0, is trivial. So let us assume n > 1.Step 1: We first show the vanishing for i �= n. The vanishing for i > n is clear.By Lemma 3.5 and Lemma 3.1, the complex K•(V, Z,m) is exact, and we mayinterpret it as a resolution of the first non-trivial term. Hence Hi(X,O(mZ)) =Hi+1(X,K•>0(V, Z,m)). The vanishing for i = 0, 1 follows from the left-exactnessof the functor Γ. By Lemma 3.6, if m ≥ 0, we are done because the Koszul complexis an acyclic resolution of the first term, so its hypercohomology coincides with itscohomology after taking Γ. In general, we may consider the E1-term of the firsthypercohomology spectral sequence of K•>0(V, Z,m). By Lemma 3.6, it looks like

Hn−1(X,OX((m+ 1)Z))⊗∧1 V ∗ d1→ Hn−1(X,OX((m+ 2)Z))⊗

∧2 V ∗ d1→ · · ·0 0...

...0 0

H0(X,OX((m+ 1)Z))⊗∧1 V ∗ d1→ H0(X,OX((m+ 2)Z))⊗

∧2 V ∗ d1→ · · ·

Note, that the d1-cohomology of the bottom sequence is what we are interestedin. The spectral sequence differential

dk : Hn−1(X,OX((m+ s)Z))⊗s∧V ∗ → Hn−k(X,OX((m+ s+ k)Z))⊗

s+k∧V ∗

hits the bottom line for k = n. Thus, the leftmost term it reaches is the one with∧n+1 V ∗ which is zero. Hence the sequence degenerates at E1 and we have for0 < i < n− 1

0 = Hi(X,O(mZ)) = Hi+1(X,K•>0(V, Z,m)) = HKi+1(V, Z,m).

Step 2: Now, let’s have a look at the last two non-trivial terms in the Koszul

complex which are Γ(X,OX((n−1+m)Z))⊗k

∧n−1V ∗ → Γ(X,OX((n+m)Z))⊗k∧n V ∗. Using the identification

∧n−1 V ∗ = V ⊗k

∧n V ∗, this map is canonicallyisomorphic to the map

Γ(X,OX((n− 1 +m)Z))⊗k V → Γ(X,OX((n+m)Z))

tensored with the identity on∧n V ∗. Its cokernel is thus R(Z)(n+m)⊗k

∧n V ∗. �

Corollary 3.9. Let Z be a non-degenerate divisor on a toric variety withNewton polytope Δ. Set n = dimΔ+ 1. We have

HidΓ(K(TΔ, Z,m)) =

{0 for i �= n

R(Z)n+m ⊗k

∧n−1TΔ for i = n

Proof. We apply the log derivation map and the contraction by h which yields∧n TΔ =∧n−1 TΔ. �

132


For an injection of k-vector spaces TΔ ↪→ W we define W by the cocartesiandiagram

TΔ −−−−→ W⏐⏐�⏐⏐�

TΔ −−−−→ W

and the complex Kl(W,Z,m) := OX((l + m)Z) ⊗k

∧lW for varying l with “the

same” differential dZ . A good way to think of Kl(W,Z,m) is as being the Koszul

complex Kl(TΔ, Z,m) pulled back from some higher dimensional toric variety X inwhich X embeds equivariantly, see Lemma 3.31.

Lemma 3.10. There is a non-canonical direct sum decomposition of the complex

K•(W,Z,m) ∼=⊕b≥0

K•−b(TΔ, Z,m+ b)⊗k

b∧W/TΔ.

Proof. The inclusion TΔ ↪→ W induces a filtration of∧l W for each l which

splits as∧l W ∼=

⊕b≥0

∧l−b TΔ ⊗k

∧b W/TΔ because we are dealing with vectorspaces. The differential dZ respects this splitting going

dZ : OX ((l + m)Z) ⊗l−b∧

TΔ ⊗b∧

W/TΔ → OX((l + 1 + m)Z) ⊗l+1−b∧

TΔ ⊗b∧

W/TΔ

The result is now just a matter of identifying the terms on the left of the righttensor symbol with the complex for TΔ and using W/TΔ = W/TΔ. �

Even though the splitting of the complex is not canonical, in a sense, thesplitting on cohomology is. For a vector space T , we occasionally write

∧top T

for∧dimT T . For an inclusion of vector spaces T ↪→ U , whenever there is no

confusion with another inclusion, we write 〈∧t T 〉l for the degree l part of the

exterior algebra ideal in∧•

U generated by∧t

T . We set n = dimΔ + 1 andHK(W,Z,m) := Hi

dΓ(K(W,Z,m)).

Proposition 3.11. With the above notation, we have

HKl(W,Z,m) = R(Z)l+m ⊗k

⟨∧topTΔ

⟩l−1

= R(Z)l+m ⊗k

∧topTΔ ⊗

∧l−nW/TΔ

Proof. Using Lemma 3.10, we have the non-canonical decomposition

HKl(W,Z,m) ∼=⊕

b≥0 HKl−b(TΔ, Z,m + b) ⊗k

∧bW/TΔ. By Cor. 3.9 the only

non-zero term on the right hand side is the one where l − b = n. Hence, we have

HKl(W,Z,m) ∼= R(Z)m+n+b ⊗∧top TΔ ⊗k

∧b W/TΔ for b = l − n. It remains toprove the canonicity. Consider the canonical filtration

l∧W =

⟨ 0∧TΔ

⟩l⊃⟨ 1∧

TΔ

⟩l⊃ · · · ⊃

⟨ n∧TΔ

⟩l⊃ {0}

The desired cohomology group comes from the non-trivial bottom term, more pre-cisely, it is the cokernel of the left vertical arrow in the diagram

O((m+ l)Z)⊗⟨∧n TΔ

⟩l

O((m+ l)Z)⊗∧n TΔ ⊗

∧l−n W/TΔ�⏐⏐dZ

�⏐⏐(dZ⊗id )

O((m+ l − 1)Z)⊗⟨∧n−1 TΔ

⟩l−1

id⊗ξ←−−−−− O((m+ l − 1)Z)⊗∧n−1 TΔ ⊗

∧l−n W/TΔ

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22 HELGE RUDDAT

The bottom map ξ is the only non-canonical map. It is supposed to be a sectionof the right non-trivial map in the exact sequence

0 −→⟨ n∧

TΔ

⟩l−1

−→⟨ n−1∧

TΔ

⟩l−1

−→n−1∧

TΔ ⊗l−n∧

W/TΔ −→ 0.

Then any choice of ξ makes the diagram commute, because O((m + l − 1)Z) ⊗〈∧n TΔ〉l−1 is contained in the kernel of the left vertical map. Thus, we get a canoni-cal identification of the cokernels of the vertical maps which shows HKl(W,Z,m) =

C⊗∧n

TΔ⊗∧l−n

W/TΔ where C = coker(q) and q : Γ(X,OX((m+l−1)Z))⊗T ∗Δ →

Γ(X,OX((m + l)Z)), u ⊗ n → u · ( 1f ∂nf). We see that C = R(Z)l+m. Let ι(h)

denote the contraction by the natural projection h : TΔ � k. We can apply theisomorphisms

• ι(h) :∧n

TΔ →∧dimTΔ TΔ

• W/TΔ = W/TΔ

• 〈∧top

TΔ〉l →∧top

TΔ ⊗∧l−dimTΔ W/TΔ, αtop ∧ αW → αtop ⊗ [αW ].

to obtain the result. �

3.3. Jacobian rings and Newton polyhedra. We wish to analyze the re-lation of R(Z) to Jacobian rings in this subsection. We stay with the previousnotation and assume here that Δ is a simplex and define a relatively open subsetof its cone by

C(Δ)\/ = C(Δ)

∖ ⋃v∈Δ[0]

(v, 1) + C(Δ) .

It is easily seen to be the half open parallelepiped C(Δ)\/ = {∑

v∈Δ[0] λv(v, 1) | 0 ≤λv < 1}. For each l ∈ Z≥0 we may intersect this with the hyperplane {(m, l) |m ∈M ⊗Z R} and project to the first summand to have

Δ\l/ = {∑

v∈Δ[0]

λvv | 0 ≤ λv < 1, l =∑v

λv} ⊆ l ·Δ.

This space was already defined in [6], Ch. 9. One finds Δ\l/ = ∅ ⇔ l ≥ |Δ[0]| =dimΔ + 1. This gadget has nice functorial properties. For a subset S ⊆ Rn, letrelintS denote the relative interior of S in span

RS.

Lemma 3.12. We have

a) Δ\l/ ∩ (l · F ) = F \l/ b) Δ\l/ =∐F⊆Δ

relintF \l/

where in each of these F ⊆ Δ is supposed to be a face.

Proof. To see that a) is true just note that a face F is determined by the setof vertices it contains. It is then given by points of Δ for which λv = 0 wheneverv �∈ F . Then a) easily follows and b) is a consequence of a). �

Definition 3.13. We denote by Γ\l/(Z) the subspaces of Γ(X,OX(lZ)) gen-erated by the images of zm under the first map in Lemma 3.2 for which m ∈ Δ\l/.This is independent of a particular equation of Z.

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We consider the monoid ring k[C(Δ) ∩ (M ⊕ Z)] which is Noetherian andgraded by the second summand. Assume that we are given a homogeneous elementof degree one f =

∑m∈Δ∩M fmz(m,1). One defines the Jacobian ideal of f by

Jf = (∂nf |n ∈ Hom(M ⊕ Z,Z))

where ∂nf =∑

m∈Δ∩M 〈(m, 1), n〉fmz(m,1). Relating to Griffith’s work, Batyrevhas used the notation R0, R1 for two types of toric Jacobian rings in [2]. In [6],Borisov and Mavlyutov have picked up this notation.

Definition 3.14 (Batyrev, Borisov, Mavlyutov). We set

R0(f,Δ) = k[C(Δ) ∩ (M ⊕ Z)]/Jf .

and define R1(f,Δ) as the subspace generated by lattice points in relintC(Δ). Thelatter is a module over R0(f,Δ).

We can put a ring structure on R(Z) by writing it as the quotient of the globalsections tensor algebra Γ(X,OX(•Z)) by the ideal generated in degree one by thelinear system of log derivatives V .

Lemma 3.15. There is a graded ring isomorphism

R(Z) ∼= R0(f,Δ)

which is canonical up to a multiplicative constant.

Proof. We obtain the inverse of the desired isomorphism via the unique ring

map induced in degree one by⊕

m∈Δ∩M k · z(m,1)· 1f−→ Γ(X,OX(Z)) as given in

Lemma 3.2. It maps the respective ideals to each other as can be seen from thedefinition of the log derivation map. The remark about the multiplicative constantaddresses the fact that R0(f,Δ) = R0(af,Δ) for a ∈ k× whereas the isomorphismdepends on a. �

We define the vector space kΔ\l/∩M =

{∑m∈Δ\l/∩M amzm

∣∣ am ∈ k}.

Lemma 3.16. Let f ∈ k[C(Δ) ∩M ⊕ Z]1 be arbitrary. The map

kΔ\l/∩M → R0(f,Δ)l

given by sending zm to z(m,l) is an injection.

Proof. We call an f with the property fm �= 0 ⇔ m ∈ Δ[0] Fermat. Notethat the lemma is true for all Fermat f because then Jf = (z(v,1)|v ∈ Δ[0]) and its

degree l part is (Jf )l =

{∑m∈lΔ∩M

m�∈Δ\l/amz(m,l)

∣∣∣∣ am ∈ k

}. The kernel of the map in

the lemma gives a coherent module on the space Spec k[fm|m ∈ Δ ∩M ] of all f . Itis trivial at Fermat points and therefore also in a neighbourhood U of these points.

There is an operation of the torus Gm(k)Δ∩M on this space under which kΔ\l/∩M

is invariant. We deduce the result for all f which lie in an orbit with non-trivialintersection with U . The other f can easily be checked directly. �

We call f non-degenerate if the corresponding Z is non-degenerate.

Proposition 3.17. If f is non-denerate then, for each l, the map

kΔ\l/∩M → R0(f,Δ)l

given by sending zm to z(m,l) is an isomorphism.

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24 HELGE RUDDAT

Proof. It is not hard to see that if f is Fermat then Z is non-degenerate andwe saw in the proof of the previous lemma that for these f the assertion is true.The result then follows from Lemma 3.16 and ([2], Thm. 4.8) which states thattwo linearly equivalent non-degenerate divisors have the same graded dimensionsfor their Jacobian rings. �

A general lattice polytope can always be triangulated by elementary simplices,so they form the building blocks of lattice polytopes. A special case of these isthe standard simplex which is one that is isomorphic to the convex hull of 0 anda subset of a lattice basis. Gross and Siebert required these in [20] for the outermonodromy polytopes to make their Hodge group computation work. Here is howthese relate to Jacobian rings and thus to the cohomology of the Koszul complex.

Lemma 3.18. For a lattice simplex Δ with lattice M , the following are equiva-lent

a) Δ is standardb) relint (n ·Δ) ∩M = ∅ for n ≤ dimΔ.c) R0(f,Δ)l = 0 for l > 0 and some non-degenerate f .

Proof. Without loss of generality, we may assume that dimΔ = rankM . Setd = dimΔ. Note that b) is equivalent to relint (dΔ) ∩ M = ∅. By applying atranslation, we may assume that some v0 ∈ Δ[0] is the origin. Let v1, ..., vd be theother vertices. They form a basis if and only if there is no lattice point other than

the vertices contained in the parallelepiped P = {∑d

i=0 λivi | 0 ≤ λi ≤ 1} spannedby these vectors. Note that P ⊂ d ·Δ and

P ∩ ∂(d ·Δ) = {∑d

i=1 vi} ∪ {x ∈ P |x =∑d

i=1 λivi with some λi = 0}.

To see the implication b)⇒a) now assume a) doesn’t hold, so there is some latticepoint x =

∑λivi ∈ P which isn’t a vertex. We may assume λi > 0 by adding vi if

necessary. Now x �∈ P ∩∂(d·Δ) and therefore x ∈ relint (dΔ)∩M . We get a)⇒b) byrepeatedly subtracting vi for each 1 ≤ i ≤ d from an arbitrary x ∈ relint (dΔ) ∩Mas long as the result x′ is still contained in dΔ. We have x′ ∈ P and x′ isn’t avertex of P .

By Prop. 3.17, c) is equivalent to

Δ\l/ ∩M = 0 for each l > 0.

By the same argument as before, Δ is non-standard if and only if there is x′ =∑di=1 λivi ∈ M with 0 ≤ λi < 1 and some λi > 0. For such an x′, set I = {i |λi > 0}

and let F be the face of Δ which is

F =

{the convex hull of {vi, i ∈ I} if

∑i∈I λi ∈ N

the convex hull of 0 and {vi, i ∈ I} otherwise

Let l be the smallest integer greater or equal to∑

i∈I λi and λ0 = l −∑

i∈I λi.

We find x′ ∈ F \l/ and by Lemma 3.12 a) we have x′ ∈ Δ\l/ which proves c)⇒a).

Using Lemma 3.12 a) again, the converse becomes clear because x =∑d

i=1 λivi ∈relint (F \l/) for some l > 0 and some face F ⊆ Δ yields an element x+

∑i:λi=0 vi ∈

relint (dΔ), i.e., b)⇒c). �

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3.4. The Koszul complex and log differential forms. We use the no-tation and setting from the previous section, i.e., we have a toric variety X anda non-degenerate semi-ample Cartier divisor Z with Newton polytope Δ. Let Dhere, unlike in the previous section, denote the boundary divisor of X, i.e., thecomplement of the big torus. For a normal variety Y with an effective Cartierdivisor E we denote by Ωr

Y/k(log (E)) the sheaf of differential r-forms with at most

logarithmic poles along E. In general this doesn’t need to be a coherent sheaf. Inour situation dealing with a toric boundary divisor, however, this will be the case.For an OY -module F , as usual, we set F(E) = F ⊗OY

OY (E). Note that there isa canonical isomorphism

ΩrX/k(log (D)) = OX ⊗Z

r∧M

by mapping u⊗m1 ∧ ... ∧mr on the right to u · dzm1

zm1∧ ... ∧ dzmr

zmr on the left.

Lemma 3.19. For each r there is an exact sequence

0 → ΩrX/k(log (D + Z))(−Z) → Ωr

X/k(logD)res−→ Ωr

Z/k(log (D ∩ Z)) → 0

Proof. The first two non-zero terms are naturally contained in the sheaf ofrational forms. We check that the first injects in the second. Let g be a functionon an open chart which is invertible outside D ∪ Z. By the local irreducibility ofZ, we may assume that on that chart Z is irreducible. Let f be a local equationof Z. Either g is invertible outside D or g = g′ · fk with g′ invertible outside D.Assume the latter, then

f · dgg

= f · fkdg′ + kg′fk−1df

fkg′=

fdg′ + kg′df

g′=

fdg′

g′+ kdf

is a form with at most logarithmic poles along D, so the first non-trivial map is well-defined and injective. We also see that Ω•

X/k(log (D + Z))(−Z) is the subalgebra

of Ω•X/k(logD) locally generated by f and df . This is the kernel of the surjection

to Ω•Z/k(log (D ∩ Z)) and we are done. �

Remark 3.20. a) If E is a normal crossings divisor on a complex man-ifold Y , then Hk(Y \E,C) = Hk(Y,Ω•

Y/C(log(E))) and Hkc (Y \E,C) =

Hk(Y,Ω•Y/C(log(E))(−E)) whereHc is cohomology with compact support.

We have, so to say, combined these two concepts.b) One can show that if Z is non-degenerate, we have a toroidal pair (X,D+

Z) in the sense of [8]. For such a toroidal pair (Y,E), Danilov definesΩ•

(Y,E) as the kernel of Ω•Y/k →

⊕E′ Ω•

E′/k where the sum ranges over the

irreducible components of E. The author calls these modules differentialforms with logarithmic zeros. For k = C, he then shows the degenerationof the hypercohomology spectral sequence of Ω•

(Y,E) at the E1 term (see

[9]). In [2], Batyrev uses the Poincare dual Ω•Y (log(E)). Our constructions

reside somewhere in between these two and are determined to be used aslocal contributions to the Hodge data of toric Calabi-Yau degenerations.

For each rationally generated subspace T ⊆ M ⊗Z k, in other words, for eachsaturated Z-submodule T ∩M of M , we can view OX ⊗k

∧rT as a free submodule

of ΩrX/k(logD) which we wish to call T ∩ Ωr

X/k(logD). As long as we make sure

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26 HELGE RUDDAT

that locally df ∈ T ∩ Ω•X/k(logD), i.e., TΔ ⊆ T , we obtain by the previous lemma

an induced exact sequence

0 → T ∩ΩrX/k(log (D+Z))(−Z) → T ∩Ωr

X/k(logD)res−→ T ∩Ωr

Z/k(log (D∩Z)) → 0.

This sequence will be of most interest to us in the case where T = TΔ. Leth : TΔ → k denote the canonical projection as before. Given TΔ ↪→ W , we alsodefine it for W . There is an exact sequence

0 → W → Wh−→ k → 0.

Definition 3.21. Given an inclusion TΔ ↪→ W , we define a map

πr : Kr(W,Z,−r − 1) → OX ⊗r∧W

by the composition Kr(W,Z,−r − 1)dZ−→ Kr+1(W,Z,−(r + 1)) = OX ⊗k

∧r+1W

id⊗ι(h)−→ OX ⊗Z

∧r W where ι(h) means contraction by h. We mostly write π for

πr. That the image lies in∧r

W inside∧r

W can be seen by applying ι(h) to theexact sequence

0 →r+1∧

W →r+1∧

W → W/W ⊗r∧W → 0.

Lemma 3.22. Let f be an equation of Z. Using Lemma 3.2, the map πr isexplicitly given by

u → u ·∑

m∈Δ∩M

fm1

fzm for r = 0

u⊗ (v ∧ α) → u ·∑

m∈Δ∩M

fm1

fzm ⊗ ((v −m) ∧ α) for r > 0

where α ∈∧r−1W and v = (v, 1) for a vertex v ∈ Δ.

Proof. The map dZ was already given explicitly in Lemma 3.4. This lemmadirectly follows by composing with ι(h). �

For the next theorem we need the following elementary lemma

Lemma 3.23. a) For a vector space V , a subspace T of codimension one,

v ∈ V \T , α ∈∧r−1

V and v ∧ α ∈∧r

T , we have v ∧ α = 0b) For a vector space V , a subspace T of codimension k, v1, ..., vk ∈ V

which are linearly independent modulo T and α1, ..., αk ∈∧r−1

V we have∑ki=1 vi ∧ αi ∈

∧r T ⇒∑k

i=1 vi ∧ αi = 0

Proof. It is clear that b) implies a), so we only need to prove b). Becausev1, ..., vk descend to a basis of V/T , they induce a splitting V ∼= T ⊕ V/T . This in

turn induces an isomorphism of graded algebras∧• V ∼=

⊕j≥0

∧•−j T⊗∧j V/T. By

construction vi ∈∧0

T⊗∧1

V/T so we have∑k

i=1 vi∧αi ∈⊕

j≥1

∧r−jT⊗

∧jV/T.

On the other hand∧r

T =⊕

j=0

∧r−jT ⊗

∧jV/T . This implies the assertion. �

The following theorem is going to tell us that the Koszul complex resolvesthe sheaves TΔ ∩ Ωr

X/k(log (D + Z))(−Z). Later, we will be using this resolution

to compute the cohomology of these specific log differential forms which are thebuilding blocks of the log Hodge sheaves on a h.t. toric log Calabi Yau space as wewill to see in the next section.

138


Theorem 3.24. For each r there is an exact sequence

0 −→ K0(TΔ, Z,−(r + 1)) −→ K1(TΔ, Z,−(r + 1)) −→

... −→ Kr(TΔ, Z,−(r + 1))π−→ TΔ ∩ Ωr

X/k(logD)res−→ TΔ ∩ Ωr

Z/k(log (D ∩ Z)) −→ 0.

Proof. The exactness of the first part follows by the overall non-degeneracyhypothesis, Lemma 3.5 and Lemma 3.1. Moreover, the sequence is exact atKr(TΔ, Z,−(r+1)) if and only if ι(h) is injective on the image of dZ : Kr(TΔ, Z,−(r+

1)) → Kr+1(TΔ, Z,−(r+1)). Note that Kr+1(TΔ, Z,−(r+1)) = OX ⊗∧r TΔ. We

have kern ι(h) = OX ⊗∧r

TΔ. We may consider the map dZ over the field of frac-tions Quot (OX) in which OX(lZ) canonically embeds for each l. The advantage isthat via Lemma 3.4 the map dZ is then given by wedging with an element dF ofQuot (OX)⊗ TΔ which is not in Quot (OX)⊗ TΔ. Applying Lemma 3.23, a) yields

dF ∧ α ∈ OX ⊗r∧TΔ ⇔ α = 0.

This finishes showing the exactness at Kr(TΔ, Z,−(r + 1)). By Lemma 3.19, theonly thing left to prove is that

im (πr) = TΔ ∩ ΩrX/k(log (D + Z))(−Z).

We take a look at the image of πr in a toric chart. Let σ be a maximal cone inthe fan of X and U = Spec k[Pσ] the corresponding chart where Pσ = σ∨ ∩ M .There is a unique vertex v ∈ Δ such that Δ − v ⊂ σ∨ (otherwise σ∨ would haveto contain a straight line which is impossible for a full-dimensional σ). Choose thestandard local trivialization OX(−Z)|U ∼= OU such that the section 1

f zm is given

by zm−v. Let us first consider the case r = 0. By Lemma 3.22, the map π0|Ubecomes multiplication with

∑m∈Δ∩M fmzm−v which is just an equation of Z on

U . Thus, the case r = 0 reduces to the standard sequence

0 → OX(−Z) → OX → OZ → 0

which is exact. Now assume r > 0. By Lemma 3.22, the map πr becomes

πr|U : OU ⊗k

∧r TΔ → OU ⊗k

∧r TΔ

u⊗ w ∧ α → u ·∑

m∈Δ∩M fmzm−v ⊗ (w −m) ∧ α.

In other words, the image of πr|U is the degree r part of the exterior algebra idealin OU ⊗

∧•TΔ generated by

Lw :=∑

m∈Δ∩M

fmzm−v ⊗ (w −m) for w ∈ Δ[0].

We set fU :=∑

m∈Δ∩M fmzm−v. As we already mentioned, this defines Z ∩U . Wehave

Lv =∑

m∈Δ∩M fmzm−v ⊗ (m− v)

=∑

m∈Δ∩M fmzm−v dzm−v

zm−v in ΩX/k(logD)= dfU ,

Lw − Lv =∑

m∈Δ∩M fmzm−v ⊗ ((m− w)− (m− v))=

∑m∈Δ∩M fmzm−v ⊗ (v − w)

= fU ⊗ (v − w).

The set {v−w |w ∈ Δ[0]} spans TΔ. With the same argument as at the end of theproof of Lemma 3.19 where we described TΔ∩Ωr

X/k(log (D+Z))(−Z) as the exterior

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28 HELGE RUDDAT

algebra generated by fU and dfU , we see that im (πr) = TΔ∩ΩrX/k(log (D+Z))(−Z),

and we are done. �Corollary 3.25. For each r ∈ N≥0, there is an exact sequence

0 −→ K0(W,Z,−(r + 1)) −→ K1(W,Z,−(r + 1)) −→

... −→ Kr(W,Z,−(r + 1))π−→ OX ⊗

r∧W

res−→ coker (π) −→ 0.

Proof. This follows from Lemma 3.10 and Theorem 3.24. �We wish to give the generalized Ωr

X/k(log (D + Z))(−Z) a functional name.

Definition 3.26. Given an inclusion TΔ ↪→ W we define Cr(W,Z) by the exactsequence

Kr−1(W,Z,−(r + 1))dZ−→ Kr(W,Z,−(r + 1)) → Cr(W,Z) → 0.

By Corollary 3.25, we have

Cr(W,Z) = im (πr).

Proposition 3.27. For each r, there is an acyclic resolution

0 → Cr(W,Z) → Kr+1(W,Z,−(r + 1)) → ... → KdimW (W,Z,−(r + 1)) → 0

which is functorial in W .

Proof. The exactness, once again, follows from the overall non-degeneracyhypothesis, Lemma 3.5 and Lemma 3.1. Acyclicity follows from Lemma 3.6. For

TΔ ↪→ W ′, a TΔ-map W → W ′ clearly induces a map∧l

W →∧l

W ′ for each land a map of complexes K•(W,Z,−(r + 1)) → K•(W ′, Z,−(r + 1)). �

Corollary 3.28. If Z �= ∅, we have for all r, p

Hp(X, Cr(W,Z)) = HKr+p+1(W,Z,−(r + 1)) = R(Z)p ⊗k 〈top∧

TΔ〉r+p,

in particular, Hp(X, Cr(TΔ, Z)) = 0 for p+ r �= dimΔ.

Proof. For p = 0 this uses Γ(Kr(W,Z,−(r+1))) = Γ(OX(−Z)⊗k

∧r W ) = 0,and it follows from Prop. 3.27 and Prop. 3.11. �

Remark 3.29. We may naturally extend the de Rham differential on log formsby the unique derivation extending the following map on monomial functions

d : OX ⊗k

∧r W → OX ⊗k

∧r+1Wzm ⊗ α → zm ⊗m ∧ α.

This map is compatible with the inclusions of Cr(W,Z) by Theorem 3.24. In par-ticular d is trivial on “constant differential forms”, i.e., those where m = 0.

Remark 3.30 (Moving Δ). Recall that we have fixed a translation repre-sentative of the lattice polytope Δ which is the Newton polytope of the non-degenerate divisor Z. We want to discuss now what happens if we move Δ. SetTR = span

R{v − w | v, w ∈ Δ[0]} Assume that we have two embeddings p1 : Δ ↪→

TR, p2 : Δ ↪→ TR which differ by a translation by some integral vector v ∈ TR, i.e.,p2 = p1 + v. For any TΔ ↪→ W , this induces an automorphism

Sv : W → W , w → w + ι(h)(w) · v

140


which maps the cone over p1(Δ) to the cone over p2(Δ). Moreover, it induces anisomorphism of complexes

K•p1(W,Z, r)

∧• Sv−→ K•p2(W,Z, r)

where we use the indices p1, p2 to denote by which translation representative thecomplex is constructed. To see this, consider Lemma 3.4. This isomorphism co-incides with taking the detour via V ∗, i.e., Sv = ∂∗

Z,p2◦ (∂∗

Z,p1)−1, see the proof

of Lemma 3.4. An important point is that the map πl commutes with Sv whichfollows from Lemma 3.22. The cohomology of K• is invariant under Sv because∧top

TΔ is. Having said all this, whenever K• comes up in this paper, we keep theposition of the polytope arbitrary and just need to make sure that all additionalconstructions commute with translations of Δ.

Lemma 3.31. Let Δ′ be a face of Δ and X ′ the corresponding toric subvariety ofX. We assume that Z is non-degenerate and so Z ′ = Z∩X ′ is also non-degenerate.There is a canonical isomorphism of sequences on X ′

... → Kr(TΔ, Z,−(r + 1))|X′π−→ TΔ ∩ Ωr

X/k(logD)|X′res� TΔ ∩ Ωr

Z/k(log (D ∩ Z))|X′↓ ↓ ↓

... → Kr(TΔ, Z′,−(r + 1))π−→ OX′ ⊗

∧r TΔres� coker (π)

where the bottom sequence comes from TΔ′ ↪→ TΔ. More generally, given TΔ′ ↪→W , we have an analogous isomorphism K•(W,Z,−(r + 1))|X′ → K•(W,Z ′,−(r +1)).

Proof. By Remark 3.30 we can move Δ in the unique position such that Δ′

embeds in it as the face corresponding to the stratum X ′ in X. Let f be an equationof Z then f ′ = f |X′ is an equation of Z ′ on X ′. The vertical isomorphisms areinduced by OX(lZ)|X′ = OX′(lZ ′) for varying l. The lemma becomes clear afterchecking the behaviour of the differential after restriction to X ′. The differential inthe upper row is

u⊗ α → u ·∑

m∈Δ∩M

fm(f−1zm)|X′ ⊗ (m, 1) ∧ α.

Since f−1zm|X′ = 0 for m �∈ Δ′ and f−1zm|X′ = (f ′)−1zm for m ∈ Δ′, this is just

u⊗ α → u ·∑

m∈Δ′∩M

fm(f ′)−1zm ⊗ (m, 1) ∧ α.

which is the differential in the lower row. �

4. The cohomology of C (Ωr)

4.1. An acyclic resolution on a stratum. In this section we wish to applythe constructions of section 3. The key object will be Cr(...) for various parameters.The overall hypothesis is now that X is a h.t. toric log CY space. This implies that,for each τ , there is only one Zτ , Δτ ,Δτ ,Ωτ , Rτ , respectively. We will implicitlyuse the following lemma a lot.

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30 HELGE RUDDAT

Lemma 4.1. Let e : τ1 → τ2 in P. The following are equivalent.

i) We ∩Δ �= ∅ ii) e ∈ Δiii) Zτ1 ∩Xτ2 �= ∅ iv) Zτ1 ∩Xτ2 = Zτ2

v) There is some h ∈ Ωτ2 which factors through e.vi) There is some h ∈ Rτ1 which factors through e.

vii) {ω → τ1e→ τ2 → ρ |ω ∈ P [1], ρ ∈ P [dimB−1], κωρ �= 0} �= ∅

Proof. iii) and iv) are equivalent by the h.t. property. ii)⇒ i) is trivial.The inverse direction is clear if τ1 �= τ2 because e is the only edge of nontrivialintersection with We. If τ1 = τ2, then e is a point which is contained in everyedge of Δ ∩We. The equivalence of ii) and vii) follows from the description of Δand v)⇔ vi) ⇔ vii) is trivial. It remains to show that ii) is equivalent to iii). IfZτ1 = ∅ then the negation of vii) follows. On the other hand, vii) implies Zτ1 �= ∅via Lemma 2.1, so we may assume Zτ1 �= ∅. Because Zτ1 ⊆ Xτ1 is a divisor notcontaining any toric stratum and Xτ2 ⊆ Xτ1 is a stratum, we have

Zτ1 ∩Xτ2 �= ∅ ⇔ Zτ1 ∩ Int (Xτ2) �= ∅.

By the assumption Zτ1 �= ∅ there exists P [1] � ω0h→ τ1 which represents an edge

contained in Δ. By Lemma 2.1,

e ◦ h ∈ Δ ⇔ Zω0∩Xτ2 �= ∅ ⇔ Zω0

∩Xτ1 ∩Xτ2 �= ∅ ⇔ Zτ1 ∩Xτ2 �= ∅.

We therefore need to show that e ∈ Δ ⇔ e◦h ∈ Δ but this is just ii)⇔v) whichwe have shown already. �

Remark 4.2. Note that possibly Zτ1 , Zτ2 �= ∅ but Zτ1 ∩Xτ2 = ∅. This happensif we have a diagram

ω1 �� τ1e

��

��

�� ρ1

ω2 �� τ2 �� ρ2

where none of the maps factors through any other map and κω1ρ1, κω2ρ2

> 0.In other words, the points id τ1 and id τ2 might be contained in Δ but not theconnecting edge.

Definition 4.3. Let M be a lattice, σ some lattice polytope in M ⊗Z R, Σ itsnormal fan. Let ψ be a piecewise linear function with respect to Σ coming fromanother lattice polytope Δσ ⊆ M ⊗Z R in the sense of Lemma 2.5. Let τ ⊆ σ be aface and τ ∈ Σ the corresponding cone. We can restrict ψ to the star of τ [see [15],p.52] and obtain a piecewise linear function on the normal fan of τ which comesfrom a face of Δσ. We denote this face by

Δσ ∩ τ

because it is the intersection of Δσ with a translate of the tangent space of τ .

Definition 4.4. For any e : τ1 → τ2, we define Ze = Zτ1 ∩Xτ2 and polytopes

Δe = Δτ1 ∩ τ2 = Newton (Ze) andΔe = Δτ2 ∩ τ1.

142


It is not hard to see that we get a similar statement as in Lemma 2.4. Thereare surjections

{ edges of Δe} {ω → τ1e→ τ2 → ρ |κωρ �= 0}�� { edges of Δe}.

In particular, Δe = {0} if and only if e �∈ Δ. The following lemma is a directconsequence of Lemma 4.1 and the definition of h.t..

Lemma 4.5. For e : τ1 → τ2, we have

Ze,Δe, Δe =

{Zτ2 ,Δτ1 , Δτ2 if e ∈ Δ∅, {0}, {0} if e �∈ Δ.

By the naturality of Prop. 3.27, we have, for each r and P [0] � vg→ τ1

e→ τ2,a commutative diagram of OXτ2

-modules

(4.1)

Cr(Δ⊥e , Ze)

π−−−−→ OXτ2⊗k

∧r Δ⊥e⏐⏐�

⏐⏐�Cr(Λv,k, Ze)

π−−−−→ OXτ2⊗k

∧r Λv,k

where the vertical maps are injections. Using Lemma 3.10 to decompose the Cr’sinto Cr−s(TΔe

, Ze)’s, one finds that the diagram is cartesian. We thus have

Lemma 4.6. The sequence

0 → Cr(Δ⊥e , Ze) → Cr(Λv,k, Ze)⊕OXτ2

⊗k

r∧Δ⊥

e → OXτ2⊗k

r∧Λv,k

is exact where the first non-trivial map is (Cr(Δ⊥e ↪→ Λv),−π) and the second is

π + id ⊗∧r(Δ⊥

e ↪→ Λv).

Adapting Prop. 2.10 to the h.t. situation, we get

(Fs(e)∗Ωr

τ1)/Tors =⋂

w �=Vert (v)w∈Δτ1

kern

(Fs(e ◦ g)∗Ωr

v

ι(∂w−Vert (v))|(Ze)†−→ Ωr−1(Ze)†/k†

).

We are going to use the canonical identification Ωrv = Ωr

Xv(logDv) = OXv

⊗k∧rΛv,k as given in [20], Lemma 3.12. Pulling back differentials simplifies to re-

stricting functions and thus

Fs(e ◦ g)∗Ωrv = Fs(e ◦ g)∗OXv

⊗k

r∧Λv,k = OXτ2

⊗k

r∧Λv,k.

A choice of splitting Λv,k∼= TΔe

⊕Δ⊥e /TΔe

⊕ Λv,k/Δ⊥e induces an isomorphism

r∧Λv,k

∼=⊕a,b≥0

a∧TΔe

⊗k

r−a−b∧Δ⊥

e /TΔe⊗k

b∧Λv,k/Δ

⊥e .

This induces a decomposition

Fs(e ◦ g)∗Ωrv∼=

⊕a,b≥0

(TΔe

∩ ΩaXτ2

/k(logDτ2))⊗k

r−a−b∧Δ⊥

e /TΔe⊗k

b∧Λv,k/Δ

⊥e .

143

32 HELGE RUDDAT

Proposition 4.7. Given a choice of splitting as above, the image of the inclu-sion of (Fs(e)

∗Ωrτ1)/Tors in Fs(e ◦ g)∗Ωr

v inherits a decomposition as

⊕a,b≥0

Wa,b

where

Wa,b = OXτ2⊗k

a∧TΔe

⊗k

r−a−b∧Δ⊥

e /TΔe⊗k

b∧Λv,k/Δ

⊥e

for b = 0 and

Wa,b = kern (res a)⊗k

r−a−b∧Δ⊥

e /TΔe⊗k

b∧Λv,k/Δ

⊥e

for b > 0 with res a : TΔe∩ Ωa

Xτ2/k(logDτ2) → TΔe

∩ ΩaZe/k

(log (Ze ∩Dτ2)).

Proof. Note that the assertion is trivial if e �∈ Δ because then, by Lemma 4.5,Ze = ∅, Δe = {0},Δ⊥

e = Λv and so only the component with b = 0 contributes. Letus now assume that e ∈ Δ which implies Ze = Zτ2 , Δe = Δτ1 , Δe = Δτ2 usingLemma 4.5. We are going to show the existence of such a decomposition first. Thisis a consequence of Prop. 2.10 together with the following two observations

(1) For each w �= Vert (v), ι(∂w−Vert (v)) respects the decomposition in thesense of being the identity on the first two tensor factors when writtendown as

(TΔτ2

∩ ΩaXτ2

/k(logDτ2))⊗k

r−a−b∧Δ⊥

τ1/TΔτ2⊗k

b∧Λv,k/Δ

⊥τ1

−→(TΔτ2

∩ ΩaXτ2

/k(logDτ2))⊗k

r−a−b∧Δ⊥

τ1/TΔτ2⊗k

b−1∧Λv,k/Δ

⊥τ1 .

(2) The restriction to Zτ2 respects the decomposition by being res a on thefirst and the identity of the last two tensor factors being written as

(TΔτ2

∩ ΩaXτ2

/k(logDτ2))⊗k

r−a−b∧Δ⊥

τ1/TΔτ2⊗k

b∧Λv,k/Δ

⊥τ1

−→(TΔτ2

∩ ΩaZτ2

/k(log (Zτ2 ∩Dτ2)))⊗k

r−a−b∧Δ⊥

τ1/TΔτ2⊗k

b∧Λv,k/Δ

⊥τ1 .

Note that {w −Vert (v) |w �= Vert (v), w ∈ Δ[0]τ1 } generates TΔτ1

and therefore

⋂w �=Vert (v)

w∈Δτ1

kern

( r∧Λv,k

ι(w−Vert (v))−→r−1∧

Λv,k

)=

r∧Δ⊥

τ1

which implies the assertion for the b = 0 case.On the other hand if α is a form in a component of some a, b with b > 0

then there is a w such that ι(∂w−v)α �= 0. For α to be in Wa,b, we must have thatι(∂w−v)α restricts to 0 under res . This, however, is equivalent to α itself restrictingto 0 under res . This finishes the proof. �

The following proposition adds to Lemma 4.6.

144


Proposition 4.8. Given e : τ1 → τ2, there is a split exact sequence

0 → Cr(Δ⊥e , Ze) → Cr(Λv,k, Ze)⊕OXτ2

⊗r∧Δ⊥

e → Fs(e)∗Ωr

τ1/Tors → 0

where the last non-trivial map depends on e ◦ g : v → τ2. It induces

Fs(e)∗Ωr

τ1/Tors ∼= OXτ2⊗

r∧Δ⊥

e ⊕(Cr(Λv,k, Ze)/Cr(Δ⊥

e , Ze)).

Proof. From Lemma 4.6, we know that the beginning is exact and just needto care about the last term. As in the proof of Prop. 4.7 the assertion is trivialfor e �∈ Δ. In the other case, we have Ze = Zτ2 , Δe = Δτ1 , Δe = Δτ2 . Choosea splitting Λv,k

∼= TΔτ2⊕ Δ⊥

τ1/TΔτ2⊕ Λv,k/Δ

⊥τ1 . We are going to use Prop. 4.7.

Given its notation, all we need to show is that

im (π + id ⊗r∧F (g)∗) =

⊕a,b≥0

Wa,b.

By Lemma 3.10, the entire sequence splits up in a, b-components. For the compo-nents with b = 0 the assertion is obvious because

⊕a≥0b=0

Wa,b∼= OXτ2

⊗k

r∧Δ⊥

τ1

which clearly coincides with the image. For b > 0 we have by Prop. 4.7

Wa,b = kern (res )⊗k

r−a−b∧Δ⊥

τ1/TΔτ2⊗k

b∧Λv,k/Δ

⊥τ1

and by Theorem 3.24, kern (res a) = im (πa) = Ca(TΔτ2, Zτ2). Now, the assertion

becomes clear by writing down the a, b-decomposition of Cr(Λv,k, Zτ2) which is

Cr(Λv,k, Zτ2)∼= Ca(TΔτ2

, Zτ2)⊗k

r−a−b∧Δ⊥

τ1/TΔτ2⊗k

b∧Λv,k/Δ

⊥τ1 .

�

By Prop. 3.27, the exact sequence of the previous proposition yields an exactsequence of complexes where the right column is defined as the cokernel sequence.

.

.

.

.

.

.

.

.

.↑ ↑ ↑

Kr+2(Δ⊥e , Ze,−(r + 1)) ↪→ Kr+2(Λv,k, Ze,−(r + 1)) � Q1(Fs(e)

∗Ωrτ1

, g)

↑ ↑ ↑Kr+1(Δ⊥

e , Ze,−(r + 1)) ↪→ Kr+1(Λv,k, Ze,−(r + 1)) ⊕ OXτ2⊗

∧r Δ⊥e � Q0(Fs(e)

∗Ωrτ1

, g)

↑ ↑ = ↑Cr(Δ⊥

e , Ze) ↪→ Cr(Λv,k, Ze) ⊕ OXτ2⊗

∧r Δ⊥e � Fs(e)

∗Ωrτ1

/Tors

↑ ↑ ↑0 0 0

Proposition 4.9. We have an acyclic resolution

0 → Fs(e)∗Ωr

τ1/Tors → Q•(Fs(e)∗Ωr

τ1 , g)

Proof. This directly follows from Prop. 3.27 and Lemma 3.10. �

145

34 HELGE RUDDAT

4.2. Independence of the vertex. In this section we wish to show that thepreviously built resolution Q•(Fs(e)

∗Ωrτ1, g) doesn’t depend on g in the sense that

for another g′ there is a natural commutative diagram

Fs(e)∗Ωr

τ1/Tors

Q0(Fs(e)∗Ωr

τ1, g′)

Q0(Fs(e)∗Ωr

τ1, g)

Q0(Fs(e)∗Ωr

τ1, g′)

Q1(Fs(e)∗Ωr

τ1, g)

. . .

. . .

Before we go on, recall from Lemmma 2.9 that g : v → τ1 induces a ver-

tex Vert (g) ∈ Δ[0]τ1 . We will sometimes also denote it by Vert (v). Since Δe ∈

{Δτ1 , {0}}, we may also understand this as Vert (g) ∈ Δ[0]e . There is a dual version

as well. The data h : τ → σ ∈ P [dimB] determines a maximal cone Kσ in the fanΣτ , see ([19], Def. 1.35), on which Δτ defines a piecewise linear function, so we candefine

Vert (h) ∈ Δ[0]τ

such that Δτ −Vert (h) ⊆ K∨σ . We also use this as Vert (h) ∈ Δ

[0]e .

Let us assume we have e : τ1 → τ2, two vertex embeddings vgv→ τ1, w

gw→ τ1and h : τ2 → σh an embedding in a maximal cell. Set mh = Vert (h) and mh =

(mh, 1) ∈ TΔe. We define an isomorphism

φhgv,gw : k · mh ⊕ Λv,k → k · mh ⊕ Λw,k

as follows. Let γh be some path from v to w through the interior of σ and Tγh:

Λv,k → Λw,k be the isomorphism induced by parallel transport along γh. We set

φhgv,gw |Λv,k

(m) = Tγh(m) + 〈m,Vert (gw)−Vert (gv)〉 · mh, φh

gv,gw(mh) = mh.

Lemma 4.10. Let vgv→ τ1, w

gw→ τ1 be two vertex embeddings. The isomorphismφgv,gw defined by the commutative diagram

Λv,kφgv,gw−−−−→ Λw,k∥∥∥

∥∥∥

kmh ⊕ Λv,k

φhgv,gw−−−−→ kmh ⊕ Λw,k

is independent of the choice of h : τ2 → σh.

Proof. Step 1: The case where v, w are connected by an edge ω.Let o : ω → τ2 be the embedding of the edge in τ1 composed with e : τ1 → τ2. Leth′ : τ2 → σh′ be another inclusion in a maximal cell and γh′ a path from v to wthrough the interior of σh′ . We have

Tγh= Tγh′ ◦ T−1

γh′ ◦ Tγh.

Note that T−1γh′ ◦ Tγh

= Tγh◦γ−1

h′is a monodromy transformation along the loop

γh ◦ γ−1h′ based at v which is by ([19], section 1.5) (choosing dω to point from v to

w) given as

Tγh◦γ−1

h′(m) = (Th◦o,h′◦o

ω )∗(m)

= m+ 〈m,Vert (gw)−Vert (gv)〉 · nh◦o,h′◦oω

= m+ 〈m,Vert (gw)−Vert (gv)〉(Vert (h′)−Vert (h))= m+ 〈m,Vert (gw)−Vert (gv)〉(mh′ − mh)

146


Using this and that Vert (h′)− Vert (h) ∈ τ⊥1 is invariant under local monodromy,we get

φhgv,gw |Λv,k

(m) = Tγh(m)− 〈m,Vert (gw)−Vert (gv)〉mh

= Tγh′ ◦ Tγh◦γ−1

h′(m)− 〈m,Vert (gw)−Vert (gv)〉mh

= Tγh′ (m+ 〈m,Vert (gw)−Vert (gv)〉(mh′ − mh))+ 〈m,Vert (gw)−Vert (gv)〉mh

= Tγh′ (m) + 〈m,Vert (gw)−Vert (gv)〉mh′

= φh′

gv,gw |Λv,k(m)

Note that φhgv,gw(mh) = mh for all h : τ2 → σh ∈ P [dimB] because mh − mh is

monodromy invariant and mh is fixed by definition.

Step 2: Chains of edges.Pick some h : τ2 → σh and let ω1, ..., ωk be a chain of edges of τ1 connecting vertex vto vertex w. Let v−ωi

, v+ωibe the vertices of ωi, s.t. v = v−ω1

, v+ωi= v−ωi+1

and v+ωk= w.

Let gv−ωi, gv+

ωidenote the respective embeddings in τ1. We set φh

ωi:= φh

gv−ωi

,gv+ωi

for

each i and claimφhgv,gw = φh

ωk◦ · · · ◦ φh

ω1

Note that φhgv,gw(mh) = φh

ωk◦ · · · ◦φh

ω1(mh). Let γ

hωi

be a path through the interior

of σh connecting v−ωiand v+ωi

, then γh ∼ γhωk

◦ · · · ◦ γhω1. We compute

φhωk

◦ · · · ◦ φhω1(m) = Tγh

ωk(...(Tγh

ω1(m) + 〈m,Vert (v+ω1

)−Vert (v−ω1)〉mh) + ...)

+ 〈Tγhωk−1

◦ · · · ◦ Tγhω1(m),Vert (v+ωk

)−Vert (v−ωk)〉mh

= Tγhωk

◦ · · · ◦ Tγhω1(m) + 〈m,Vert (v+ω1

)−Vert (v−ω1)〉mh + ...

+ 〈m,Vert (v+ωk)−Vert (v−ωk

)〉mh

= Tγh(m) + 〈m,Vert (v+ωk)−Vert (v−ω1

)〉mh

= φhgv,gw(m)

where we have used that 〈Tγ(m),Vert (v1)−Vert (v2)〉 = 〈m,Vert (v1)−Vert (v2)〉holds for each path γ connecting some v1, v2 ∈ τ

[0]1 through σh.

Step 3: Combining Step 1 and 2.Let ω1, ..., ωk be a chain of edges of τ connecting v to w as described in Step 2. Weconclude

φhgv,gw

Step 2= φh

ωk◦ · · · ◦ φh

ω1

Step 1= φh′

ωk◦ · · · ◦ φh′

ω1

Step 2= φh′

gv,gw.

�Lemma 4.11 (Changing v). Let v

gv→ τ1, wgw→ τ1 be two vertex embeddings. We

have a commutative diagram

. . .

. . . . . .

. . .

id ⊗Vr φg,g′ id ⊗

Vr+1 φg,g′Fs(e)∗Ωr

τ2

/Tors

Fs(e ◦ gv)∗Ωrv

Cr(Λv, Ze)

Cr(Λw, Ze)

Fs(e ◦ gw)∗Ωrw

Kr(Λw, Ze,−(r + 1)) Kr+1(Λw, Ze,−(r + 1))

Kr+1(Λv, Ze,−(r + 1))Kr(Λv, Ze,−(r + 1))

147

36 HELGE RUDDAT

and thus a canonical isomorphism of Q•(Fs(e)∗Ωr

τ1 , g) and Q•(Fs(e)∗Ωr

τ1 , g′) as

desired at the beginning of this section.

Proof. Note that the outer rectangle clearly commutes. The only interestingnew information is, in fact, the subdiagram consisting of the five left-most terms.We are going to use the comparison of the two outgoing arrows of Fs(e)

∗Ωrτ1/Tors

which was given in [20], Lemma 3.13. We may assume that v and w are connectedby an edge ω because any two vertices can always be connected by a chain of edgesand, having proved the edge version, we have a chain of commutative diagramsinducing commutativity of the first and the last. Let o : ω → τ1 be this edge. Wechoose some h : τ2 → σh ∈ P [dimB] which determines a chart Uσh

of Xτ2 on whichwe show the commutativity of the diagram. Let f be an equation of Ze◦o, i.e., f isconstant if e ◦ o �∈ Ωτ2 and is an equation of Zτ2 otherwise. We also assume that

Vert (h) ∈ Δ[0]τ2 lies in the origin, such that f is a regular function on Uσh

.Let γh be some path from v to w through the interior of σh giving the identi-

fication Tγhof Λv and Λw which we also identify with a fixed lattice M . Note that

in loc.cit. this is denoted N and note further that we are only interested in the /k†

case. We denote the field of fractions construction by Quot . By loc.cit., the mapΓω : Fs(e ◦ gv)∗Ωr

v → Fs(e ◦ gw)∗Ωrw written as a map

Quot (Fs(e ◦ gv)∗OXv)⊗Z

r∧M → Quot (Fs(e ◦ gv)∗OXw

)⊗Z

r∧M

on the toric chart Uσhdetermined by σh is given by

Γω(1⊗ α) = 1⊗ α+df

f∧ (ι(dω)α)

where f = faω is giving the log structure at ω and dω denotes the primitive vectorpointing from v to w. Using aωdω = Vert (v)−Vert (w) whenever df �= 0, we obtain

Γω(1⊗ α) = 1⊗ α+df

f∧ (ι(Vert (v)− Vert (w))α).

The question now becomes whether

OUσh(−Zτ2)⊗Z

∧r(M ⊕ Zmh)

π−−−−→ C(M,Zτ2)|Uσh↪→ QuotOUσh

⊗Z

∧rM

φ := id⊗∧r φh

gv,gw

⏐⏐�⏐⏐�Γω

OUσh(−Zτ2)⊗Z

∧r(M ⊕ Zmh)

π−−−−→ C(M,Zτ2)|Uσh↪→ QuotOUσh

⊗Z

∧rM

commutes. Recall from the proof of Theorem 3.24, that for a suitable trivializationof OXτ2

(−Ze◦o)|Uσh, α ∈

∧r−1 M and β ∈∧r M we have

π|Uσh(1⊗ mh ∧ α) = df ∧ α π|Uσh

(1⊗ β) = f ⊗ β.

It follows

(Γω ◦ π)(1⊗ mh ∧ α) = df ∧ α+ dff ∧ ι(Vert (w)−Vert (v))(df ∧ α)

= df ∧ α+ dff ∧ df ∧ (ι(Vert (w)−Vert (v))α)

= df ∧ α

(Γω ◦ π)(1⊗ β) = f ⊗ β + dff ∧ (f ⊗ ι(Vert (w)− Vert (v))β)

= f ⊗ β + df ∧ (ι(Vert (w)−Vert (v))β)

148


The map φhgv,gw |M reads m → m + ι(Vert (w) − Vert (v))m · mh which ex-

tends in this form to M ⊕ Zmh by setting ι(Vert (w)−Vert (v))mh = 0. A simplecomputation then shows for ε ∈

∧r(M ⊕ Zmh)

r∧φhgv,gw : ε → ε+ mh ∧ ι(Vert (w)−Vert (v))ε.

We obtain

(π ◦ φ)(1⊗ mh ∧ α) = π(1⊗ mh ∧ α+ mh ∧ ι(Vert (w)−Vert (v))(mh ∧ α))= df ∧ α+ π(mh ∧ mh ∧ −ι(Vert (w)−Vert (v))α)= df ∧ α,

(π ◦ φ)(1⊗ β) = π(1⊗ β) + π(1⊗ mh ∧ ι(Vert (w)−Vert (v))β)= f ⊗ β + df ∧ (ι(Vert (w)−Vert (v))β).

We have shown Γω ◦ π = φ ◦ π and thus the above diagram commutes. We arriveat the last part of the assertion. Note that Δ⊥

e is invariant under monodromy andφgv,gw |Δ⊥

eis, in this sense, the identity. Looking at the definition of Q•, we see that

the only term affected by changing the vertex is K•(Λv,k, Ze,−(r + 1)). It is nothard to see now that φgv,gw yields the claimed isomorphism of the Q•’s. �

Definition 4.12. We use the notation Φg,g′ for the just constructed isomor-phism

Φg,g′ : Q•(Fs(e)∗Ωr

τ1 , g) → Q•(Fs(e)∗Ωr

τ1 , g′).

By the results of this section, from now on, we will sometimes use the notationQ•(Fs(e)

∗Ωrτ1) for the resolution of Fs(e)

∗Ωrτ1/Tors and only specify/choose some

g when necessary for computations.

Remark 4.13. (1) To pick up the discussion from Rem. 3.30, note thatthe span of C(Δe) is invariant under monodromy and fixed by φgv,gw . The

map φgv,gw depends on the position of Δe. It is not hard to see, however,

that moving Δe commutes with φgv,gw .

(2) The main point of φgv,gw is that the projection h : Λv,k → k doesn’tcommute with φgv,gw if Δe is non-trivial. In fact, each vertex of Δe givesone such projection. If we dualise, the projections turn into inclusionsof rays. This fits in with the construction of (B,P) from a polytope asdescribed in [19], Ex. 1.18. What we have produced here is some sort of alocal version of this. One can show that this yields a local system of rankdimB + 1 along the discriminant locus Δ. If X comes from the Batyrevconstruction, this local system is the restriction of the constant sheaf onB induced from the embedding into the surrounding vector space.

4.3. Cohomology on a single stratum. As before, we assume throughoutthat X is a h.t. toric log CY space. In this section, we compute the cohomology ofa summand of the complex C •(Ωr). We first need a lemma on locally monodromyinvariant forms. Recall that i : B\Δ ↪→ B denotes the inclusion of the non-singularlocus of B.

Lemma 4.14. Given τ1e→ τ2, the space Γ(We, i∗

∧r Λ ⊗Z k) is generated by∧r Δ⊥e and 〈

∧top TΔe〉r.

149

38 HELGE RUDDAT

Proof. Given any point y ∈ We\Δ, we may identify Γ(We, i∗∧r Λ⊗Z k) with

the subspace of∧r

Λy,k ⊗Z k of forms invariant under monodromy transformations

by loops in We\Δ. If e �∈ Δ, we have Δ⊥e = Λy and the assertion is trivial. Let

us assume e ∈ Δ. Recall from ([19], Section 1.5) that the group of monodromytransformations is generated by

α → α± κωρ · (ι(dω)α) ∧ dρ

where dω is a primitive integral vector parallel to some ω ∈ P [1] and dρ is a

primitive integral vector in ρ⊥ for some ρ ∈ P [dimB−1] such that there is an edgee : ω → ρ with e ∈ Δ which factors through e (otherwise κωρ = 0). By Lemma 2.4,

such a dω is parallel to an edge of Δe and dρ is parallel to an edge of Δe. It now

becomes obvious that∧r Δ⊥

e and 〈∧top TΔe

〉r are contained in Γ(We, i∗∧r Λ⊗Zk).

Now assume β �∈ 〈∧top TΔe

〉r +∧r Δ⊥

e . We will exhibit some monodromytransformation which doesn’t fix β. We choose ω1, ..., ωm such that dω1

, ..., dωm

form a basis of TΔeand such that there is some ρ ∈ P [dimB−1] with κωiρ �= 0 for

1 ≤ i ≤ m. Similarly we choose ρ1, ..., ρn such that dρ1, ..., dρn

form a basis of TΔe

and such that there is some ω ∈ P [1] with κωρi�= 0 for 1 ≤ i ≤ n. By Lemma 4.1,

we have κωiρj�= 0 for all i, j. We may complement dρ1

, ..., dρnto a basis B of

Λy,k by adding in particular vectors d∗ω1, ..., d∗ωn

with the property ι(dωi)b = 0 for

b ∈ B\{d∗ωi}. The basis B of Λy,k induces a basis

∧rB of

∧rΛy,k. We represent

β in this basis. Note that∧r Δ⊥

e and 〈∧top TΔe

〉r are both generated by a subset

of∧r

B. Thus, by assumption, β has a non-zero coefficient for some basis elementb in

∧rB which is not contained in these subsets. Therefore, there is some i such

that ι(dωi)b �= 0 and there is some j such that dρj

∧ b �= 0. We claim that themonodromy transformation α → α ± κωiρj

· (ι(dωi)α) ∧ dρj

changes β. This isequivalent to saying (ι(dωi

)β) ∧ dρj�= 0. This follows from (ι(dωi

)b) ∧ dρj�= 0 and

a linear independence argument. �

Theorem 4.15. For vg→ τ1

e→ τ2 with v ∈ P [0], we have

Hp(Xτ2 , F (e)∗Ωrτ1/Tors) =

⎧⎪⎪⎨⎪⎪⎩

Γ(We, i∗∧r Λ⊗Z k) for p = 0,

R(Ze)p ⊗k

〈∧top TΔe

〉 ∩∧r+p Λv,k

〈∧top

TΔe〉 ∩

∧r+pΔ⊥

e

for p > 0.

Remark 4.16. A close look makes it apparent that this representation is inde-pendent of the choice of v, resp. g, because different choices of local edge connectingpaths induce the same isomorphisms.

Proof. If e �∈ Δ, we have, by Lemma 4.1, Ze = ∅. This means R(Ze)• = 0and Fs(e)

∗Ωrτ1 = OXτ2

⊗k

∧r Λv,k so the assertion is true. We now assume e ∈ Δ

and get Ze = Zτ2 ,Δe = Δτ1 , Δe = Δτ2 . We apply the functor Hp to the diagram(4.1) to obtain

Hp(Cr(Δ⊥τ1 , Zτ2))

Hp(π)−−−−→ Hp(OXτ2⊗k

∧rΔ⊥

τ1)⏐⏐�⏐⏐�

Hp(Cr(Λv,k, Zτ2))Hp(π)−−−−→ Hp(Fs(e)

∗Ωrτ1/Tors).

150


Note that the corresponding sequence on cohomology splits because the originalsequence splits. Let’s consider the case p = 0 first. We can read off the exactsequence

0 →r∧Δ⊥

τ1 → H0(Fs(e)∗Ωr

τ1/Tors) → H0(Cr(Λv,k, Zτ2))/H0(Cr(Δ⊥

τ1 , Zτ2)) → 0.

We may use Cor. 3.28 to obtain H0(Cr(Λv,k, Zτ2)) = R(Zτ2)0 ⊗k 〈∧top TΔτ2

〉∩∧r Λv,k and H0(Cr(Δ⊥

τ1 , Zτ2)) = R(Zτ2)0 ⊗k 〈∧top TΔτ2

〉 ∩∧r Δ⊥

τ1 . We have

R(Zτ2)0 = Γ(OXτ2) = k, so the exact sequence reads

0 →r∧Δ⊥

τ1 → H0(Fs(e)∗Ωr

τ1/Tors) →〈∧top

TΔτ2〉 ∩

∧rΛv,k

〈∧top

TΔτ2〉 ∩

∧rΔ⊥

τ1

→ 0.

Therefore, H0(Fs(e)∗Ωr

τ1/Tors) is identified with the subspace of∧r

Λv,k which is

generated by∧r Δ⊥

τ1 and 〈∧top TΔτ2

〉 ∩∧r Λv,k. By Lemma 4.14, the assertion for

p = 0 follows.The case where p > 0 is even simpler because Hp(OXτ2

⊗k

∧rΔ⊥

τ1) = 0. Againusing Cor. 3.28, we directly have the assertion. �

Proof of Theorem 1.6,a). Note that the functors Γ(We, ·) and∧r+p com-

mute on presheaves of vectors spaces on We. We choose some g : v → τ1 withv ∈ P [0]. The assertion follows from Thm. 4.15 if we show that the cokernelsC1, C2 in the diagram

〈∧top TΔe

〉 ∩∧r+p Δ⊥

e� � ��

� �

��

〈∧top TΔe

〉 ∩∧r+p Λv,k

��

��

C1

��

Γ(We,∧r+p i∗Λ⊗ k)

� � �� Γ(We, i∗∧r+p Λ⊗ k) �� C2

are isomorphic. The natural map C1 → C2 is injective because the left square iscartesian which follows from Δ⊥

e = Γ(We, i∗Λ⊗Z k). Surjectivity is a consequenceof the fact that the term in the middle of the bottom row is generated by the imagesof the two incoming arrows which we know by Lemma 4.14. �

4.4. The strata combining differential. Up to now, we have been workingon a single stratum Xτ2 only. Now we take into consideration the barycentricdifferential dbct. We are going to produce an acyclic resolution of the complexC •(Ωr) to have an explicit description of the hypercohomology spectral sequenceof C •(Ωr) for the proof of Thm. 1.6,b). Our overall hypothesis is that X is a h.t.toric log CY space. Recall that the etale locally closed embedding of the stratumto X is denoted by qτ : Xτ → X. For the first half of this section, we fix a chain ofmaps

vg→

e︷︸︸︷σ1 → τ1

e→ τ2 → σ2 .

with v ∈ P [0]. We denote the composition of the first two maps by g : v → τ1. Itis not hard to see that Ze = Ze ∩Xσ2

, Δe = Δe ∩ σ1, Δe = Δe ∩ σ⊥2 . Recall from

([20], Prop. 3.8) that we get a map

qτ2,∗F (e)∗Ωrτ1/Tors → qσ2,∗F (e)∗Ωr

σ1/Tors

which factors through the restriction to Xσ2.

151

40 HELGE RUDDAT

Lemma 4.17. We have a commutative diagram

0 →qτ2,∗Cr(Δ⊥e , Ze)→qτ2,∗Cr(Λv,k, Ze) ⊕ qτ2,∗OXτ2

⊗∧r Δ⊥

e →qτ2,∗Fs(e)∗Ωr

τ1/Tors→ 0

⏐⏐�

⏐⏐�

⏐⏐�

0 →qσ2,∗Cr(Δ⊥e , Ze)→qσ2,∗Cr(Λv,k, Ze) ⊕ qσ2,∗OXσ2

⊗∧r Δ⊥

e →qσ2,∗Fs(e)∗Ωr

σ1/Tors→ 0

where the rows are given by Prop 4.8, the right vertical map is the one justmentioned and the left two vertical maps are the composition of the restriction toXσ2

, the map induced by Δ⊥e → Δ⊥

e and Lemma 3.31.

Proof. Replacing the right most non-trivial terms with

Fs(τ2 → σ2)∗ : qτ2,∗OXτ2

⊗k

r∧Λv,k → qσ2,∗OXσ2

⊗k

r∧Λv,k

clearly gives a commutative diagram by the functoriality from Prop. 3.27 and thefact that Fs(τ2 → σ2)

∗ is a functor. The assertion then follows from the commuta-tive diagram

Fs(e)∗Ωr

τ1/Tors −−−−→ Fs(e ◦ g)∗Ωrv = OXτ2

⊗k

∧rΛv,k

Fs(τ2→σ2)∗⏐⏐�

⏐⏐�Fs(τ2→σ2)∗

Fs(e)∗Ωr

σ1/Tors −−−−→ Fs(e ◦ g)∗Ωr

v = OXσ2⊗k

∧rΛv,k

and the way of construction of the sequence in Prop. 4.8. �

For exactly the same reasons, we also obtain a diagram of resolutions, replacingCr’s with the suitable Kr+1+•’s and removing the summands qτ2,∗OXτ2

⊗∧r

Δ⊥e if

• > 0. We call this map on the Q•’s

dee : qτ2,∗Q•(Fs(e)

∗Ωrτ1 , g) → qσ2,∗Q

•(Fs(e)∗Ωr

σ1, g).

We make use of the statement of Theorem 4.15 in the following Lemma.

Lemma 4.18. The map Hp(X, qτ2,∗Fs(e)∗Ωr

τ1) → Hp(X, qσ2,∗Fs(e)∗Ωr

σ1) is, for

p = 0, the restriction

Γ(We, i∗

r∧Λ⊗Z k) → Γ(We, i∗

r∧Λ⊗Z k)

induced by the canonical isomorphism

Γ(We, i∗

r∧Λ⊗Z k) = Γ(We ∩We, i∗

r∧Λ⊗Z k)

and the inclusion of open sets We ∩We ⊆ We. It is, for p > 0, induced by

Fs(τ2 → σ2)∗ : R(Ze)• � R(Ze)•,

Δe ↪→ Δe and hence 〈∧top

TΔe〉 ↪→ 〈

∧topTΔe

〉,Δe ↪→ Δe and hence Δ⊥

e ↪→ Δ⊥e .

Proof. The isomorphism Γ(We, i∗∧r

Λ ⊗Z k) = Γ(We ∩ We, i∗∧r

Λ ⊗Z k)follows from the fact that e and e can be joined by a simplex in Δ with codimensiontwo in B. Such a simplex is given by a chain τ0 � ... � τdimB−2 with τ0 ∈P [1], τdimB−2 ∈ P [dimB−1] which is a refinement of e and e. Thus by the proof of([19], Lemma 5.5), (We ∩We)\Δ is a deformation retract of We\Δ.

152


The assertion follows from computing the map dee on the Q•’s and is straight-forward. We just discuss the map of the R’s. We set F = Fs(τ2 → σ2). For each lthe natural adjunction map becomes

a : Γ(Xτ2 ,OXτ2(pZe)) → Γ(Xτ2 , F∗F

∗OXτ2(pZe)) = Γ(Xτ2 , F∗OXτ2

(pZe))

because F ∗Ze = Ze. Let Ve, Ve denote the linear systems via the log derivationmap for Ze, Ze, respectively, as given after Lemma 3.2. We may assume thatΔe is embedded such that the embedding of Δe is induced by restriction to thecorresponding face. If f is an equation of Ze, then F ∗f is an equation of Ze. Weget a map Ve → Ve by the diagram

(Nτ2 ⊕ Z)⊗Z k∂Ze−−−−→ Γ(Xτ2 ,OXτ2

(Ze))⏐⏐�⏐⏐�a

(Nσ2⊕ Z)⊗Z k

∂Ze−−−−→ Γ(Xτ2 , F∗OXτ2(Ze)).

The map on the R’s then is the cokernel of the diagram

Γ(Xτ2 ,OXτ2((p− 1)Ze))⊗k Ve −−−−→ Γ(Xτ2 ,OXτ2

(pZe))⏐⏐�⏐⏐�

Γ(Xτ2 , F∗OXσ2((p− 1)Ze))⊗k Ve −−−−→ Γ(Xτ2 , F∗OXσ2

(pZe)).

Note that the right vertical map is surjective because by Lemma 3.2 it can be

described by kl·Δe∩MXτ2 � k

l·Δe∩MXτ2 , zm → 0 if m �∈ l · Δe and zm → zm

otherwise. �

Recall the Gross-Siebert resolution from Def. 1.4. For each r, we are going toconstruct an acyclic resolution of C •(Ωr). For each τ ∈ P, choose some gτ : vτ → τwith vτ ∈ P [0]. We define the double complex

Qk,l(Ωr) =⊕

τ0 → . . . → τk︸︷︷︸e

qτk,∗Ql(Fs(e)

∗Ωrτ0/Tors, gτ0)

where the differential in the l-direction is the usual one on Q• which we are goingto denote by δ. The differential in the k-direction is

(dbct(α))τ0→...→τk+1= d

τ0→τk+1τ1→τk+1 ◦ Φgτ1 ,(τ0→τ1)◦gτ0 (ατ1→...→τk+1

)

+∑k

i=1(−1)i id (ατ0→...τi→...→τk+1)

+(−1)k+1dτ0→τk+1τ0→τk (ατ0→...→τk).

We collect some results in the following lemma.

Lemma 4.19. On the space X, we have for each r a double complex of Γ-acyclicsheaves

Q•,•(Ωr)

which is exact in both directions except at the respective first non-trivial terms. Wehave the augmentation

0 → C •(Ωr) → Q•,0(Ωr).

153

42 HELGE RUDDAT

Proof. The exactness of the augmentation and acyclicity are the content ofProp. 4.9. The commutativity of differentials reduces to Lemma 4.17 and whatwas mentioned afterwards. We are going to prove the exactness of dbct. We set

Q0k=

∧r+1 Λvτ0,k⊕

∧r Δ⊥τ0∧r+1 Δ⊥

τ0

. Recall that, for e : τ0 → τk,

Ql(Fs(e)∗Ωr

τ0/Tors, gτ0) =

⎧⎨⎩

OXτk⊗Q0

kfor l = 0

OXτk(lZτk)⊗

∧r+l+1 Λvτ0,k∧r+l+1 Δ⊥

τ0

for l > 0.

What we want to prove is a local issue. Let p ∈ X be some geometric pointand τ ∈ P be such that p ∈ Int (Xτ ). For e : τ0 → τk, we have

qτk,∗Ql(Fs(e)

∗Ωrτ0/Tors, gτ0)p = 0 if there is no τk → τ.

By Lemma 6.1, we are done if we show criterion (L) from Section 6.1. We matchthe notation by setting Ξ = τ and M(τ0,τk) = qτk,∗Q

l(Fs(e)∗Ωr

τ0/Tors, gτ0)p. Wemay fix some τ0, τk−1 ⊆ τ with τ0 ⊆ τk−1. Let (fe)e ∈

⊕τk�τk−1

M(τ0,τk) be a

compatible collection. We want to show that it lifts as required in criterion (L)from Section 6.1.

The case p ∈ Zτ0 : This implies Zτ = Zτ0 ∩Xτ and thus Δe = Δτ0 for each e : τ0 →τ ′ with τ ′ ⊆ τ . We claim that each M(τ0,τk) is the pullback of M(τ0,τk−1), i.e., forF = F (τk−1 → τk), the map dτ0→τk

τ0→τk−1induces

Ql((F ∗Fs(e)∗Ωr

τ0)/Tors, gτ0)p = F ∗Ql(Fs(e)∗Ωr

τ0/Tors, gτ0)p.Indeed, both F ∗M(τ0,τk−1) and M(τ0,τk) are⎧⎨

⎩OXτk

,p ⊗Q0k

for l = 0

OXτk(lZτk)p ⊗

∧r+l+1 Λvτ0,k∧r+l+1 Δ⊥

τ0

for l > 0.

Now criterion (L) follows from the fact that M(τ0,τk−1) is locally free on Xτk−1and

that we can always lift functions from subvarieties.

The case p �∈ Zτ0 : We set U = {τk ⊆ τ | τk � τk−1, Zτ0 ∩ Xτk �= ∅}. Consider thefollowing diagram with exact rows and columns.

K� �

��

K� �

��

∧r Δ⊥τ0

�� Q0k

��

ι(h)

��

∧r+1 Λvτ0,k∧r+1 Δ⊥

τ0

ι(h)

��∧r Δ⊥τ0

��∧r Λvτ0,k

��

��

∧r Λvτ0,k∧r Δ⊥τ0

We choose a splitting as indicated by the dashed arrow. We claim that thecompatible collection decomposes as

(fe)e = (f1e , f

2e )e ∈

⊕τk�τk−1

(M1(τ0,τk)

⊕M2(τ0,τk)

)

154


where

M1(τ0,τk)

=

{OXτk

,p ⊗∧r+l

Λvτ0 ,kl = 0

0 otherwise

and

M2(τ0,τk)

=

⎧⎪⎪⎨⎪⎪⎩

OXτk,p ⊗K l = 0, τk ∈ U

OXτk,p ⊗

∧r+l+1 Λvτ0,k

∧r+l+1 Δ⊥τ0

l > 0, τk ∈ U

0 otherwise.

Indeed, we can identify OXτk(lZτk)p = OXτk

,p and there is only one obvious way to

decompose using the map ι(h) and the chosen splitting above. One can now showthat both (f1

e )e and (f2e )e lift. The reason is again that functions from subvarieties

lift. For (f2e )e, one uses that {Xτk | τk ∈ U} is a set of strata closed under inter-

section and then the functions f2e actually glue to a function on the corresponding

subspace. We have shown (L) and can apply Lemma 6.1. �

As a corollary, we obtain a different proof of Lemma 1.5 still using the sameargument for the first term as in ([20], Thm. 3.5) though.

The finiteness of the dimensions of global sections and the computability ofcohomology is the major strength of Q•,•. The downside, however, is that itis impossible to extend the de Rham differential to it to obtain a triple com-plex. Roughly speaking, differentiating elements of OXτ2

(lZτ2) yields somethingin OXτ2

((l+ 1)Zτ2) whereas for compatibility it would have to stay in OXτ2(lZτ2).

We will later make use of the fact that exterior differentiation can at least be definedon Q•,0(Ω•).

4.5. Degeneration at E2. We are now going to prove Thm. 1.6, b). The keyingredient is the previously constructed double complex Q•,•(Ωr) We assume nowthat each Δe is a simplex in order to be able to use techniques from Section 3.3.

Definition 4.20. For each r, we define the subcomplex Q•,•top(Ω

r) ⊆ Q•,•(Ωr)by setting

Qk,ltop(Ω

r) =⊕

τ0 → . . . → τk︸︷︷︸

e

qτk,∗OXτk(lZe)⊗k

(〈top∧

TΔe〉∩

r+l+1∧ Λvτ0 ,kmod

r+l+1∧Δ⊥

e

)

for l > 0 and Qk,0top(Ω

r) = 0.

Note that the differential δ is trivial on Q•,•top(Ω

r). So to see that it is asubcomplex, we just check closedness under dbct. This follows from the closed-ness under the change of vertex operator Φ and under dee. The latter is because

〈∧top

TΔe〉 ⊆ 〈

∧topTΔe

〉.

Definition 4.21. We define the subcomplex Q•,\•/top (Ωr) of Γ(X,Q•,•

top(Ωr))

by replacing each Γ(Xτk ,OXτk(lZe)) in Γ(X,Qk,l

top(Ωr)) by Γ\l/(Ze). Again, δ is

trivial, so we have to show closedness under dbct. For this, we need to showthat the image of Γ\l/(Ze) under the restriction map dee : Γ(Xτk ,OXτk

(lZe)) →Γ(Xτk+1

,OXτk+1(lZe)) is contained in Γ\l/(Ze). The Newton polytope of lZe is

a face of the Newton polytope of lZe, so this follows from Lemma 3.12 and thedefinition of Γ\l/(Ze).

155

44 HELGE RUDDAT

Lemma 4.22. Assume that X is a Fermat toric log CY space. For k ≥ 0, l > 0,we have an injection

Qk,\l/top (Ωr) ↪→ H l

δΓ(X,Qk,•(Ωr)).

Proof. This follows from Thm. 4.15, Prop. 3.17 and Lemma 3.15. �Lemma 4.23. Let

⊕p,q K

p,q be a double complex with differentials d′, d′′ which

is bounded in p and q, denote by D = d′ + (−1)pd′′ the differential on the totalcomplex Tot•(K•,•). Assume the following criterion:

For each x ∈ Kp,q with d′′x = 0 and d′x = d′′yfor some y ∈ Kp+1,q−1, there is somez ∈ Kp,q−1 such that d′(x+ d′′z) = 0.

0

→

xd′→ d′′y

� →

z y

Then, its first spectral sequence degenerates at

Ep,q2 : Hp

d′Hqd′′(K

•,•) ⇒ Hp+qD (Tot•(K•,•)).

Proof. Let [·]k mean taking the class in Ek. For x1 ∈ Kp,q, the image of [x1]kunder the differential dk is given by [d′(xk)]k for some zig-zag

0

x1 d′x1

x2 → d′x2

. . .. . .

→

→

→

→

→d′′

d′

xk d′xk

The criterion implies that for some representative x1 all xk for k ≥ 2 can be chosento be zero and thus dk = 0 for k ≥ 2. �

Proof of Theorem 1.6,b). We are going to apply Lemma 4.23 to the doublecomplexΓ(X,Q•,•(Ωr)). Let δ denote the differential in the second direction. Supposex ∈ Γ(X,Qk,l(Ωr)) with δx = 0 and dbctx = δy for some y ∈ Γ(X,Qk+1,l−1(Ωr)).For l = 0 we have y = 0 and there is nothing to show, so assume l > 0. ByProp. 3.7 and Prop. 3.17, changing x by adding a δ-coboundary, we may assume

that x ∈ Qk,\l/top (Ωr). Then dbctx ∈ Q

k+1,\l/top (Ωr), and the injection

Qk+1,\l/top (Ωr) ↪→ H l

δΓ(X,Qk+1,•(Ωr))

from Lemma 4.22 shows that y = 0. This establishes the hypothesis of Lemma 4.23which we may now apply. �

156


5. Mirror symmetry of stringy and affine Hodge numbers

5.1. Base change of the affine Hodge groups. Recall from ([20], Lemma3.12), for v ∈ P [0], the identification Ωr

v = OXv⊗k

∧rΛv,k. We define

Λr =⊕

v∈P[0]

(qv)∗OXv⊗k

r∧Λv,k

which becomes a complex (Λ•, d) under exterior differentiation. There is a barycen-tric resolution in the sense of Section 6.1 for this complex via

C k(Λr) =⊕

e:τ0→...→τk

⊕g:v→τ0

v∈P[0]

(qτk)∗Fs(e ◦ g)∗OXv⊗k

r∧Λv,k,

and the barycentric differential is induced by the restriction for a vertex whichfactors through the extended edge and the zero map on that vertex otherwise. Wewon’t make use of the following lemma but give it for completeness.

Lemma 5.1. For each r, we have an exact sequence

0 → Λr → C 0(Λr) → C 1(Λr) → ....

where the first non-trivial map is the obvious one.

Proof. Injectivity at the first non-trivial term is obvious. Given any τ and ageometric point x ∈ Int (Xτ ), we have C 0(Λr)x =

⊕g:v→τ0→τ

v∈P[0](qτk,∗Fs(g)

∗OXv)x⊗k∧r

Λv,k. An element of this maps to zero under dbct if and only if it is a compatible

collection which implies that it lifts to⊕

v→τ (qv,∗OXv)x ⊗k

∧r Λv,k for each vcomponentwise. The inverse is also true, so we have exactness also at the secondnon-trivial term. The exactness of the tail follows from Lemma 6.1 upon verifyingcriterion (L) which is easy. �

Lemma 5.2. Given a c.i.t. toric log CY space X, there is an acyclic resolutionI•,•,• with augmentation

0 → C •(Ω•) → I•,•,0

such that the exterior differential d is trivial on Γ(X, I•,•,0).

Proof. It suffices to construct an injective map of complexes C •(Ω•) → I•,•,0

where Ik,r,0 is acyclic for each k, r. The remainder of I•,•,• can then be addedby an injective resolution, e.g., Godement’s canonical resolution. We claim thatwe may just take Ik,r,0 = C k(Λr). By the c.i.t. hypothesis and by what we saidat the beginning of Section 2.3, namely that we have the result ([20], Prop. 3.8),i.e., for each g : v → τ0 and e : τ0 → τk, we have an inclusion Fs(e)

∗Ωr/Tors ↪→Fs(e ◦ g)∗Ωr = Fs(e ◦ g)∗OXv

⊗k

∧rΛv,k. We may use this to get the injection

C •(Ω•) ↪→ I•,•,0 termwise as

Fs(e)∗Ω•

τ0/Tors →⊕

g:v→τ0

v∈P[0]

(qτk)∗Fs(e ◦ g)∗OXv⊗k

r∧Λv,k.

Because Γ(X,C k(Λr)) consists of constant differential forms only, d is trivial on it.Acyclicity is apparent. �

157

46 HELGE RUDDAT

Proof of Theorem 1.11. We show that, for e : τ1 → τ2,

(5.1) H0(Xτ2 , F (e)∗Ωpτ1/Tors) = Γ(We, i∗

p∧Λ⊗Z k).

For the h.t. case this is part of Thm. 4.15. We can extended this to the c.i.t. caseas follows. Recall that there is a set of Cartier divisors Zτ1,1, ..., Zτ1,t which arethe reduced components of the closure of Z ∩ Int (Xτ ). We set Ze,i = Zτ1,i ∩Xτ2

which might be empty. The empty ones won’t play a role in the following, so let us

exclude them. Fix some P [0] � vg−→ τ1 and define Ωp

i by the exact sequence

0 → Ωpi → Fs(e ◦ g)∗Ωp

vδi−→ Ωp−1

(Ze,i)†/k†→ 0

where δi is the map δ in Prop. 2.10 composed with the ith projection. By loc.cit.,we then get (Fs(e)

∗Ωpτ1)/Tors =

⋂ti=1 Ω

pi . By Thm. 4.15, we have

Γ(Xτ2 ,Ωpi ) =

( p∧Λv ⊗Z k

)Gi

whereGi is the group of those local monodromy transformations which are transvec-tions that fix (Δτ1,i ∩ τ2)

⊥ and shear by a vector in T(Δτ2,i)∩τ⊥1. Because the mon-

odromy on We\Δ is generated by {Gi | 1 ≤ i ≤ t}, we have

Γ(We, i∗

p∧Λ⊗Z k) =

t⋂i=1

( p∧Λv ⊗Z k

)Gi

and conclude (5.1) by the left-exactness of the functor Γ. To prove a), note that

Hp,qlog (X) = Hq(X,Ωp) = H

q(X,C •(Ωp)).

Let J •,• be an injective resolution of C •(Ωp) with the augmentation C •(Ωp) ↪→J •,0. If we denote by D the total differential of the double complex Γ(X,J •,•)then

Hp,qlog (X) = Hq

DΓ(X,J •,•).

There is an injection

kern (D|Γ(X,J q,0))

imD ∩ Γ(X,J q,0)↪→ Hq

DΓ(X,J •,•)

The left hand side can be rewritten as Hqdbct

Γ(X,C •(Ωp)). By what we said before

this coincides with the Cech cohomology group Hq({Wτ | τ ∈ P}, i∗∧p Λ ⊗Z k)

and we are done with part a).The proof of b) is similar. Let I•,•,• be a resolution as given in Lemma 5.2

and let D′ denote the total differential on Γ(X, I•,•,•). We have Hk(X,Ω•) =Hk

D′Γ(X, I•,•,•). Because d is trivial on Γ(X, I•,•,0) by arguing as in a) we get foreach p, q with p+ q = k an injection

Hp,qaff (X) ↪→ H

k(X,Ω•).

Once again from the triviality of d on Γ(X, I•,•,0) one concludes that these injectionscan be extended to their direct sum as required in the assertion. �

Definition 5.3. For a c.i.t. toric log CY space X, we call

T p,qlog (X) = Hp,q

log (X)/Hp,qaff (X)

the log twisted sectors.

158


As explained in ([20], Cor. 3.24), by ([19], Prop. 1.50), we obtain, for a general(B,P), the following.

Theorem 5.4 (Gross, Siebert). Assume that the holonomy of B is contained inSLn(Z)�Zn where n = dimB. Let ϕ be some multi-valued strictly convex piecewiselinear function on (B,P) and (B, P, ϕ) be the discrete Legendre dual. If X is atoric log CY space with dual intersection complex (B, P) then

Hp,qaff (X) ∼= Hn−p,q

aff (X).

So we see that the affine Hodge numbers fulfill mirror symmetry duality. Theduality for the ordinary Hodge numbers follows for the case where the twistedsectors vanish. We are going to consider what happens if this is not the case.

5.2. Twisted sectors in low dimensions. We now consider the situationup to dimension 4 as defined in Theorem 1.13.

Lemma 5.5. Let Δ be an elementary simplex with dim Δ = 3. Let f benon-degenerate. For k > 0, we have R0(f, C(Δ))k = R1(f, C(Δ))k. Moreover,R1(f, C(Δ))k = 0 for k �= 2.

Proof. The natural inclusion R1(f, C(Δ))k ↪→ R0(f, C(Δ))k becomes an iso-morphism for k > 0 because each lattice point in kΔ which isn’t a sum of a latticepoint in (k− 1)Δ and one in Δ lies in the relative interior of kΔ because otherwiseit would have to be in some facet of Δ. This, however, can be excluded by the factthat each facet of Δ is is a two dimensional elementary simplex, thus a standardsimplex, and Lemma 3.18.

The second assertion then works out as follows. It is clear for k = 0. It followsfrom elementarity of Δτ and Prop. 3.17 for k = 1. The cases k = 3, 4 then followby the pairing given in [6], Prop. 6.7. �

Proof of Theorem 1.13. By Thm. 1.11, Haff :=⊕

p,q Hp,qaff injects in the

E1-term of the hypercohomology spectral sequence of Ω• and survives to the limit.Thus, kern d1 ∩Haff = 0 and im d1 ∩Haff = 0. Cases a) and c) follow if we showthat Hp,q

log �= Hp,qaff for only one pair p, q. For a) this is p = q = 1 which we deduce

from Thm. 1.6. For c) the exceptional pair is p = q = 2 which we also deduce fromThm. 1.6 together with Lemma 3.18 to see that only three-dimensional simplicescontribute to higher cohomology terms and eventually Lemma 5.5 and Thm. 1.6to locate the contribution. Similarly to show b), we demonstrate that Hp,q

log �= Hp,qaff

only for (p, q) ∈ {(1, 2), (2, 1)}. We compute the log twisted sectors via Thm. 4.15and Lemma 4.18. We keep the convention that ω’s denote one-dimensional andτ ’s two-dimensional faces. Note that R(Zω)1 = Γ\1/(Zω) contains a canonicalsubspace induced from lattice points in the relative interior of Δω which we denote

by Γ\◦1/(Zω). For e : ω → τ , Ze = Zτ , we have dimΔe = dimΔω = 1 and

dim Δe = dim Δτ = 1. Given g : v → ω, we obtain

〈∧top TΔτ

〉∩∧2 Λv,k

〈∧top TΔτ

〉∩∧2 Δ⊥

ω

∼= Λv,k/Δ⊥ω∼= k

〈∧top TΔτ

〉∩∧3 Λv,k

〈∧top TΔτ

〉∩∧3 Δ⊥

ω

∼= k

〈∧top TΔω

〉∩∧2 Λv,k

〈∧top TΔω

〉∩∧2 Δ⊥

ω

∼={

Λv,k/Δ⊥ω∼=k

1 for dim Δω=1

0 for dim Δω=2

〈∧top TΔω

〉∩∧3 Λv,k

〈∧top TΔω

〉∩∧3 Δ⊥

ω

∼= k

〈∧top TΔτ

〉∩∧2 Λv,k

〈∧top TΔτ

〉∩∧2 Δ⊥

τ

∼={

Λv,k/Δ⊥τ∼=k

1 for dimΔτ=1

Λv,k/Δ⊥τ∼=k2 for dimΔτ=2

〈∧top TΔτ

〉∩∧3 Λv,k

〈∧top TΔτ

〉∩∧3 Δ⊥

τ

∼= k.

159

48 HELGE RUDDAT

We consider the differential d1 on cohomology degree q = 1, 2 of the E1-term of thehypercohomology spectral sequence of C •(Ωp) for p = 1, 2.

q = 1, p = 1 :(⊕

τ∈P[2] R(Zτ )1 ⊗ Λv,k/Δ⊥τ

)⊕(⊕

ω∈P[1]

dim Δω=1

R(Zω)1 ⊗ Λv,k/Δ⊥ω

)

−→⊕

ω→τ R(Zτ )1 ⊗ Λv,k/Δ⊥ω

q = 1, p = 2 :(⊕

ω∈P[1] R(Zω)1)⊕(⊕

τ∈P[2] R(Zτ )1)−→

⊕ω→τ R(Zτ )1

q = 2, p = 1 :⊕

ω∈P[1] R(Zω)2 −→ 0q = 2, p = 2 : 0 −→ 0

where the sum on the right is only over edges ω → τ which are contained in Δ. Ifwe show that the first map is injective and the second surjective, we are done.

We show the surjectivity of the second map. We can rewrite the map as⊕ω Γ\

◦1/(Zω)⊕

⊕K V 0

K →⊕

K V 1K where K runs over the connected components

of Δ\Δ0. We show that V 0K → V 1

K is surjective for each K. Note that both spacesare a direct sum of spaces isomorphic to R(ZτK )1 for a suitable τK in K. It is nothard to see that V 0

K → V 1K is isomorphic to the Cech complex of a locally constant

sheaf on K with fibre kdimR(ZτK

)1 . The contractibility of K therefore implies thedesired surjectivity.

A similar argument works for the injectivity of the first map by quasi-isomor-phically projecting it to

⊕ω∈P[1]

dim Δω=1

R(Zω)1⊗Λv,k/Δ⊥ω −→

⊕τ∈P[2]

R(Zτ )1⊗coker

(Λv,k/Δ

⊥τ ↪→

⊕ω→τ

Λv,k/Δ⊥ω

)

and identifying this map with⊕

K W 0K →

⊕K W 1

K for suitable W 0K , W 1

K , each of

which is isomorphic to the dual of a Cech complex of a locally constant sheaf withfibre k

dimR(ZτK)1 on K. �

Note that we only needed the weaker criterion of contractibility of those com-ponents K of Δ\Δ0 where dimR(ZτK )1 > 0. On the other hand, if this is notgiven for one K and the locally constant sheaves constructed in the proof haveglobal sections on K, we have T 1,1

log (X) �= 0 �= T 2,2log (X).

Corollary 5.6. For the cases considered in Theorem 1.13, at most the fol-lowing log twisted sectors are non-trivial

a) T 1,1log (X) ∼=

⊕ω∈P[1] R(Zω)1

b) T 1,2log (X) ∼=

⊕ω∈P[1] R(Zω)2 ⊕

⊕K R(ZτK )1 and

T 2,1log (X) ∼=

⊕ω∈P[1] Γ\

◦1/(Zω)⊕

⊕K R(ZτK )1

c) T 2,2log (X) ∼=

⊕τ∈P[2] R(Zτ )2

Note that that in b) R(Zω)2 ∼= Γ\◦1/(Zω). It is expected that the Picard-

Lefschetz operator maps T 2,1log (X) isomorphically to T 1,2

log (X).

Proof of Theorem 1.15. Part a) is the combination of Cor. 1.8 and The-orem 1.13. To prove part b), note that the general fibre Xt has isolated sin-gularities in these cases. Each singularity is described by a local model as re-ferred to in Prop. 2.8. See also ([20], Prop. 2.2). The degeneration is locally

160


Spec k[K∨∩(Mτ⊕Z2)] → Speck[N] whereK is the cone over (τ×{e1})∪(Δτ×{e2}).Here, τ,Δτ ⊂ Nτ ⊗ R, Nτ is a lattice of rank dimB − 1, Mτ = Hom(Nτ ,Z)and the generator of N maps to e∗1. The general fibre is thus locally given byk[C(Δτ )

∨ ∩ (Mτ ⊕ Z)]. So we have a singularity in Xt for each non-standardinner monodromy polytope Δτ . In case c), these are non-standard elementary 3-simplices. In case a), these are intervals of length greater than one.Borisov and Mavlyutov have identified a space whose dimension gives the differencehp,qst − hp,q (see [6], Def. 8.1). For each singularity this is R1(ωτ , C(Δτ )) for some

general ωτ . Under mirror symmetry, the Kahler parameter ωτ is supposed to be-come the log moduli parameter fτ . Even though we cannot make this rigorous atthe moment, we still have dimR1(ωτ , C(Δτ )) = dimR1(fτ , C(Δτ )) because an in-ner monodromy polytope of (B, P) is an outer monodromy polytope of (B,P), i.e.,Δτ = Δτ . Using Cor. 5.6, Lemma 3.15 and Lemma 5.5, we deduce the result. �

6. Appendix

6.1. Barycentric complexes. For convenience, we include here a slight mod-ification of ([19], A.1). Let Ξ be a d-dimensional polytope and Pair be the finitecategory with

objects: {(σ1, σ2) |σ1 ⊆ σ2 ⊆ Ξ are faces}morphisms: (τ1, τ2) → (σ1, σ2) for σ1 ⊆ τ1, τ2 ⊆ σ2

Let Ab denote the category of abelian groups. We assume to have a functor

Pair → Abe = (σ1, σ2) → Me.

Note that there is at most one morphism between any two objects e1, e2 in Pairwhose image under this functor we denote by ϕe1e2 . Whenever the source is clear wewill also write ϕe2 . The barycentric cochain complex (C•

bct, d•bct) associated with the

image of this functor is the complex of abelian groups Ck =⊕

σ0�σ1�...�σkM(σ0,σk)

with differentials

(dkbct(f))σ0σ1...σk+1=

k+1∑i=0

(−1)iϕ(σ0,σk+1)(fσ0...σi...σk+1)

where a means the omission of a. It is easy to check that this is a complex, i.e.,dk+1bct ◦ dkbct = 0. Assume we have some subset U of the set of objects of Pair. We

call an element (fe)e ∈⊕

e∈U Me a compatible collection if, for each e1, e2, e ∈ Uwith morphisms e1 → e, e2 → e, we have ϕefe1 = ϕefe2 . We consider the followingcriterion

(L) For each σ0 ⊆ σk−1, every compatible collection (fe)e ∈⊕

σk�σk−1M(σ0,σk)

lifts, i.e., there is some g ∈ M(σ0,σk−1) such that

f(σ0,σk) = ϕ(σ0,σk)g for each (σ0, σk).

Lemma 6.1. If (Me)e satisfies (L) then the associated barycentric complex isacyclic.

Proof. We wish to write a cocyle (fσ0...σk)σ0...σk

as a coboundary of a (k−1)-cochain (gσ0...σk−1

)σ0...σk−1. We construct the gσ0...σk−1

by descending induction on

161

50 HELGE RUDDAT

m = dim σk−1 = d+ 1, ..., 0. The induction hypothesis is that

fσ0...σk=

k∑i=0

(−1)iϕ(σ0,σk)(gσ0...σi...σk)

whenever dimσk−1 ≥ m. The base case withm = d+1 is empty because dimΞ = d.For the induction step consider some σ0 � ...σk−1 with dimσk−1 = m−1. We wantto find gσ0...σk−1

such that for any σk containing σk−1

(−1)kϕ(σ0,σk)(gσ0...σk−1) = fσ0...σk

−k−1∑i=0

(−1)iϕ(σ0,σk+1)gσ0...σi...σk.

All terms on the right hand side are known inductively. We view the right handsides for varying σk as an element of

⊕σk−1⊆σk

M(σ0,σk). If we show that this

constitutes a compatible collection, we get gσ0...σk−1from criterion (L) and are

done with the proof. So let us do this and assume we have some σk+1 containingσk−1. We need to show that

(6.1) ϕ(σ0,σk+1)

(fσ0...σk

−k−1∑i=0

(−1)iϕ(σ0,σk+1)(gσ0...σi...σk)

)

is independent of σk for σk−1 � σk � σk+1. For i ≤ k the induction hypothesisimplies

fσ0...σi...σk+1=

i−1∑j=0

(−1)jϕ(σ0,σk+1)(gσ0...σj ...σi...σk+1)

−k+1∑

j=i+1

(−1)jϕ(σ0,σk+1)(gσ0...σi...σj ...σk+1).

Plugging this into the cocycle condition

ϕ(σ0,σk+1)

(fσ0...σk

)= (−1)k

k∑i=0

(−1)ifσ0...σi...σk+1,

the first term of (6.1) gives

fσ0...σk−1σk+1(i = k) plus a sum over ϕ(σ0,σk+1)gσ0...σi...σj ...σk+1

.

For 0 ≤ i < j < k the coefficient of ϕ(σ0,σk+1)gσ0...σi...σj ...σk+1is (−1)k times

(−1)i(−(−1)j) + (−1)j(−1)i = 0. Contributions involving ϕ(σ0,σk+1)(gσ0...σi...σk)

come from the second term in (6.1) and from j = k + 1; they cancel as well. Thus(6.1) equals

fσ0...σk−1σk+1+ (−1)k

k−1∑i=0

(−1)i(−1)k(−ϕ(σ0,σk+1)gσ0...σi...σkσk+1).

This shows the claimed independence of (6.1), and hence the existence of gσ0...σk−1.�

162


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4. Batyrev, V. V.: “Stringy Hodge numbers of varieties with Gorenstein canonical singularities”,Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Sci. Publishing, RiverEdge, NJ, (1998) : p.1–32.

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6. Borisov, L. A.; Mavlyutov, Anvar R.: “String cohomology of Calabi-Yau hypersurfaces viaMirror Symmetry”, Adv. Math., 180(1), (2003) : p.355–390.

7. Borisov, L. A.; “Towards the Mirror Symmetry for Calabi-Yau Complete intersections inGorenstein Toric Fano Varieties”, preprint math.AG/9310001.

8. Danilov, V. I.: “The geometry of toric varieties”, Russ. Math. Surveys, 33, (1978) : p.97–154.9. Danilov, V. I.: “De Rham complex on toroidal variety”, Algebraic Geometry: Proc. US-USSR

Symposium, Chicago, 1989, ed. by I. Dolgachev and W. Fulton, Lecture Notes in Math., 1479,Springer-Verlag, Berlin, (1991) : p.26–38.

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Hautes Etudes Sci. Publ. Math., 35, (1968) : p.259–278.

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arxiv:0803.402123. Hartshorne, R.: “Algebraic geometry”, Graduate Texts in Mathematics, Springer-Verlag, 52,

(1977)24. Kato, F.: “Log smooth deformation theory”, Tohoku Math. J., 48(3), (1996) : p.317–354.25. Kato, K.: “Logarithmic structures of Fontaine–Illusie”, Algebraic analysis, geometry, and

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30. Mavlyutov, A. R.: “Semiample hypersurfaces in toric varieties”, Duke Math. J., 101(1),(2000) : p.85–116.

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Math. Institut, Universitat Freiburg, Eckerstr.1, 79104 Freiburg, Germany

Current address: Dep. of Mathematics, University of California in Berkeley, CA 94720 USAE-mail address: [email protected]

164

Tropical Theta characteristics

Ilia Zharkov

Abstract. This note is a follow up of [MZ07] and it is largely inspired bya beautiful description of [BN07] of non-effective degree g − 1 divisors viachip-firing game. We consider the set of all theta characteristics on a tropicalcurve and identify the Riemann constant κ as a unique non-effective one amongthem.

First we recall the basic setup and main players. For more details the readercan look in [MZ07] and references there in.

Let Γ be a connected finite graph and V1(Γ) be the set of its 1-valent vertices.We say Γ is a metric graph if the topological space Γ \ V1(Γ) is given a completemetric structure and Γ is its compactification. In particular, all leaves have infinitelengths. Introducing a new interior two-valent vertex on any edge is set to givean equivalent metric graph, and a tropical curve C is an equivalence class of suchgraphs. Its genus is g = b1(Γ) for any representative Γ.

The metric allows one to talk about affine and piece-wise linear functions on Cwith integral slopes. At every vertex v we may define the set of outward primitivetangent vectors ξi. Then any PL function f defines a principal divisor

(f) =∑p∈C

(

val(p)∑i=1

∂f(p)

∂ξi)p.

In general, a divisor D =∑

aipi is a formal linear combination of points in C withintegral coefficients. We say: D1 ∼ D2 if D1 −D2 is a principal divisor, D ≥ 0 ifall ai ≥ 0, and D has degree degD =

∑ai. The degree 2g − 2 divisor

K :=∑p∈C

(val(p)− 2)p

is called the canonical divisor.Let Aff be the sheaf of Z-affine functions (i.e., local PL functions f with (f) =

0). Define the integral cotangent local system TZ∗ on C by the following exactsequence of sheaves:

0 −→ R −→ Aff −→ TZ∗ −→ 0.

The rank g free abelian group of 1-forms ΩZ(C) on C (a.k.a. the circuit lattice) isformed by the global sections of TZ∗. Let Ω(C)∗ be the vector space of R-valuedlinear functionals on ΩZ(C). Then the integral cycles H1(C,Z) form a lattice Λ in

2010 Mathematics Subject Classification. Primary 14T99.The research is partially supported by the NSF FRG grant DMS-0854989.

1



165

2 ILIA ZHARKOV

Ω(C)∗ by integrating over them. We define the tropical Jacobian (cf. [BHN97])to be

J(C) := Ω(C)∗/H1(C,Z) ∼= Rg/Λ.

Let us fix a reference point p0 ∈ C. Given a divisor D =∑

aipi we choosepaths from p0 to pi. Integration along these paths defines a linear functional onΩZ(C):

μ(D)(ω) =∑

ai

∫ pi

p0

ω.

For another choice of paths the value of μ(D) will differ by an element in Λ. Thus,

we get a well-defined tropical analog of the Abel-Jacobi map μd : Divd(C) → J(C).The tropical Abel-Jacobi theorem (cf. [BHN97],[BN07],[MZ07]) says that

for each degree d the map μd factors through Picd(C) (the group of divisors modulolinear equivalence):

Divd(C)

μd��

�� Picd(C)

φ

��J(C)

and φ is a bijection. Both maps μd and φ depend on the base point p0 unless d = 0.Let Wg−1 ⊂ Picg−1(C) denote the Abel-Jacobi image of the set of effective divisorsof degree g − 1.

An explicit solution to the Jacobi inversion problem is provided by introducingthe tropical theta function as follows. The metric on C defines a symmetric positivebilinear form Q on Ω(C)∗ by setting Q(�, �) := length(�) on simple cycles �. That,in turn, defines a convex Λ-quasi-periodic PL function on Ω(C)∗:

(0.1) Θ(x) := maxλ∈Λ

{Q(λ, x)− 1

2Q(λ, λ)}, x ∈ Ω(C)∗,

that can be thought as a section of a polarization line bundle on J(C). Its cornerlocus defines the theta divisor [Θ] ⊂ J(C) on the Jacobian.

For λ ∈ J(C) let [Θλ] = [Θ]+λ be the translated theta divisor on J(C) and letDλ := μ∗[Θλ] denote the (effective, of degree g) pull back divisor of [Θλ] to the curvevia the Abel-Jacobi map μ : C → J(C). The Riemann constant κ ∈ Picg−1(C)(which we are after) enters into two following tropical analogs of classical theorems,either of which can serve as its definition.

Theorem 1 (Jacobi inversion, [MZ07]). There exists a universal κ ∈ Picg−1(C)such that μg(Dλ) + κ = λ for all λ ∈ J(C).

Theorem 2 (Riemann’s theta divisor [MZ07]). Wg−1+κ = [Θ], and 2(−κ) =μ2g−2(K).

Thus we see that −κ is a distinguished square root of the canonical class.

Definition 3. Divisor classes K ∈ Picg−1(C) such that 2K = μ2g−2(K) arecalled theta characteristics. We set K0 = −κ.

Lemma 4. Among the set of 2g theta characteristics K0 +12Λ/Λ ⊂ Picg−1(C)

only K0 is not in Wg−1.

166

TROPICAL THETA CHARACTERISTICS 3

Proof. According to the Theorem 2 it suffices to prove that among the two-torsion points 1

2Λ/Λ ⊂ J(C) only 0 is not in [Θ]. But since the form Q is notdegenerate the maximum at x = 0 in (0.1) is achieved by a single term, namely byλ = 0. Hence 0 ∈ Ω(C)∗ is not in the corner locus of Θ.

On the other hand if a non-zero λ ∈ 12Λ belongs to the interior of the maximal

Voronoi cell containing 0, then so does −λ (because Θ(x) is an even function). Butthis is impossible since λ− (−λ) = 2λ ∈ Λ \ {0}. �

Remark 5. The fact that one theta characteristic is distinguished in the trop-ical case is due to a choice of a Lagrangian splitting of the lattice H1(CC,Z) whendegenerating classical curves. The definition of the classical theta function dependson this choice, and once it is made there is a distinguished even element among theclassical theta characteristics (cf. e.g. [Mu83]).

Next we will give an explicit geometric description of the theta characteristics.To identify −κ, the only non-effective one, we recall from [MZ07] that given anacyclic orientation on (some graph Γ in the class of) C we can define two degreeg − 1 divisors

K± =∑p∈C

(val±(p)− 1)p

which we called moderators. Here val± stands for the number of outgoing/incomingedges. A moderator is not linearly equivalent to an effective divisor and all noneffective degree g−1 classes in Picg−1(C) comes this way ([BN07], [MZ07]). Alsonote that K+ +K− = K.

Let S ⊂ C be a finite subset. Consider the distance function dS(x) = dist(S, x).Its gradient flow defines an acyclic orientation on C. Let K±

S be the two associatedmoderators.

Lemma 6. μg−1(K±S ) = K0.

Proof. Because K+S + K−

S = K it suffices to show that K+S ∼ K−

S . In fact,

we claim that K+S = K−

S + (dS). Indeed, at any p ∈ C the slope of dS along anoutward primitive vector ξi is +1 if ξi agrees with the gradient flow of dS , and-1 if it does not. Thus, deg(dS)|p = val+(p) − val−(p), which put together givesK+

S −K−S =

∑p∈C(val+ − val−)p = (dS). �

As a consequence we note that for all subsets S the corresponding K±S represent

the same class K0. Explicitly, K−S′−K−

S = 12 (dS′−dS). The minimal representatives

(having a single pole) of K0 are given by S = q, a point in C. Then K−q is the

q-reduced form of K0 (cf. [BN07], [MZ07]).Next we identify effective theta characteristics. Let γ be a nontrivial element

in H1(C,Z/2Z) = Λ/2Λ. Let |γ| ⊂ C be the support of γ. It has only even-valentvertices. Thus one can choose a cyclic orientation on |γ|, or in other words, a simplelift (i.e., a cycle with edges of multiplicity 0 or 1) γ ∈ H1(C,Z) of γ.

Let dγ(x) := dist(|γ|, x). The gradient flow of dγ together with the orientationof γ gives an orientation on C, which, in turn, defines a pair of degree g−1 divisors

K±γ =

∑p∈C

(val±(p)− 1)p,

167

4 ILIA ZHARKOV

which depends only on γ, not on its lift γ. Since the inducing orientation is notacyclic, the K±

γ are not expected to be moderators. In fact, K−γ is manifestly

effective because the flow has no source point.As in the moderator case we have K+

γ + K−γ = K and K+

γ −K−γ = (dγ). Thus

every K±γ represents a theta characteristic. We will write Kγ for the class of K±

γ in

Picg−1(C).

Lemma 7. Kγ −K0 = 12γ in J(C).

Proof. First, note that 12γ is well defined in J(C), i.e. independent of the

choice of γ. Next we take S to be any finite subset of |γ| containing all boundarypoints of C\|γ| and all vertices of |γ|. Then |γ|\S consists of disjoint open intervals,and we denote by M the set of their mid points.

Clearly dγ = dS on C \ |γ|, hence K±γ = K±

S on C \ |γ|. On the other hand, on|γ| we have

K−γ =

∑p∈S

(1

2valγ(p)− 1)p, and K−

S = M − S,

where valγ(p) denotes the valence of p as a vertex of |γ|. Put together,

K−γ −K−

S =∑p∈S

(1

2valγ(p))p−M.

Now the orientation on |γ| specifies a system of paths starting at the points of Mand ending at the points of S. The linear functional on Ω(C) defined by thesepaths is precisely equal to 1

2 γ, and its projection to J(C) is 12γ. On the other hand,

since there are exactly 12 valγ(p) paths ending at every vertex p ∈ S this functional

represents the divisor K−γ −K−

S . �Putting all three above lemmas together we arrive at our final statement.

Theorem 8. The map γ �→ Kγ from H1(C,Z/2Z) to the set of theta charac-teristics is an isomorphism of affine Z/2Z-spaces. Among all Kγ only K0 = −κ isnot (linearly equivalent to) an effective divisor.

References

BHN97. Roland Bacher, Pierre de la Harpe, and Tatiana Nagnibeda. The lattice of integral flowsand the lattice of integral cuts on a finite graph. Bull. Soc. Math. France, 125(2):167–198,1997.

BN07. Matthew Baker and Serguei Norine. Riemann-Roch and Abel-Jacobi theory on a finitegraph. Advances in Mathematics, 215:766–788, 2007.

MZ07. Grigory Mikhalkin and Ilia Zharkov. Tropical curves, their Jacobians and Theta func-tions. http://arxiv.org/abs/math.AG/0612267v2, 2007.

Mu83. David Mumford. Tata lectures on theta. I, volume 28 of Progress in Mathematics.Birkhauser Boston Inc., Boston, MA, 1983.

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