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Hodge Theory and Moduli

Phillip Griffiths

Informal notes for a course of four lectures given in the summer school in Levico Terme (Trento) June 6–10(2016). Content based in part on work in progress with Mark Green, Radu Laza and Colleen Robles.

Institute for Advanced Study and the University of Miami

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Outline

I. Introduction; moduliI Overview of the lecturesI Moduli and singularities

II. Hodge theoryI Pure Hodge structures and their variations; period

mappingsI Mixed Hodge structures and degenerations of pure

Hodge structures

III. Hodge theory (continued); algebraic surfacesI Moduli space of polarized Hodge structures and its

Hodge-theoretic completionI H and I -surfaces

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IV. Period mapping at the boundaryI Which semi-log-canonical singularities can contribute to

non-trivial limiting mixed Hodge structures?I Realization of the Hodge-theoretic boundary strata by

the boundary of the KSBA moduli space for H and Isurfaces

I Extension of the Liu-Rollenske construction andmaximally degenerate H and I -surfaces

Selected references

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I.A. Introduction

I Lectures will be at the confluence of

Hodge theory

IIII

IIII

I

ttttttttt

Moduli/singularity

theory

algebraic surfaces

of general type

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I KSBA have constructed a quasi-projective moduli spaceM for smooth surfaces X that

I are minimal and of general type

I have given

h2,0(X ) and

h1,0(X )(Hodge theoretic)

PPK 2X (numerical-Hilbert polynomial)

and where M has a canonical completion M to aprojective variety whose points correspond to surfaces Xwhere

I X has semi-log-canonical (slc) singularities

I Essentially no examples where the structure of ∂M andthe corresponding surfaces are known (for relatedquestions see Alexeev, Hassett, Laza).

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I In joint work with Green, Laza and Robles we arestudying this question in the case when

I h2,0(X ) = 2, h1,0(X ) = 0;I K 2

X = 2, 1 (H-surface and I -surface respectively)

I Idea is to use Hodge theory as a guide to the structure of∂M. Specifically to

I use representation theory construct a canonicalHodge-theoretic completion Γ\De of the moduli spaceΓ\D of the equivalence classes of polarized Hodgestructures of weight 2;

I show that the period map Φ : M→ Γ\D captures thegeometry of X (local Torelli), and that it extends to

Φe : M→ Γ\De ;

I analyze the surfaces that map to a given component in∂(Γ\D).

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I The general objective of these lectures is to discuss thebackground and some of the work in progress on thisprogram, with emphasis on phenomena that arenon-classical in that they do not occur for curves, abelianvarieties, and K3 surfaces. Basically these arise as aconsequence of the infinitesimal period relation IPR,which appears first in weight 2 and when h2,0 = 2 whereit is a contact distribution.

I In order to have at least one specific objective for theselectures, following is a result that will emerge; thebackground and what appears in the statement will beexplained in the lectures.

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I Theorem: (i) For h2,0 = 2 the boundary ∂(Γ\D) has astratification

II

$$III

II

0 // I

<<yyyyyy

""EEE

EE IV // V

III

::uuuuu

(ii) For H and I surfaces the map Φe : ∂M→ ∂(Γ\De)realizes this diagram.

(iii) For I -surfaces, the principal components of Φ−1e (I)

will be described.

(iv) For I -surfaces, there is a unique rigid X withΦe(X ) ∈ V.

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I.B. Moduli and singularities

I Informal discussion — general references are the threeexpository papers by Kollar, Catanese, Kovacs in the“Handbook of Moduli,” Volume 25, International Press

I For moduli spaces one wants to

specify a type T

of varieties X

construct−−−−−→

MT = quasi-projective

variety whose points are

equivalence classes of X ’s

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I Even better, construct a versal family FT →MT with

X

//___ FT

B //___ MT

I Classical examples are

X =

smooth curve of genus g = 2,

polarized abelian variety,

polarized K3 surface

In these cases the period domain D parametrizing thepotential polarized Hodge structures of X ’s is a Hermitiansymmetric domain (HSD). We will refer to X ’s where Dis a HSD as the classical case.

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I In these lectures will take first non-classical case

X =

smooth algebraic surface of general type with

given K 2X , pg (X ) = h2,0, q(X ) = h1,0(X ) = 0

ww

D = SO(2a, b)/U(a)× SO(b) a = h2,0, b = h1,1

= HSD⇔ a = 1 (K3-case), we take a = 2

(non-classical cases of Hodge theory also enter in moduliof CY’s, hyperKahler varieties,. . . )

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I In moduli want canonical completion M of M; in 1st

approximation want the smooth fibres of X∗ → ∆∗ tohave unique limits Xt → X0. Done by imposing twoconditions

(i) allowable singularities of X0 (local along X0)(ii) ωX0 ample (global condition on X0).

These have to do with relative birational geometry ofX→ ∆ (modifications along X0 and base change).

I Among problems in general moduli theory where all fibresare smooth are

I jump of structure;

X→ ∆ with Xt = F2, t 6= 0

PP X0 = P1 × P1;

will be discussed in the appendix to this lecture.

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I non-separatedness

X→ ∆ smooth and where

X0 has node — Picard-Lefschetz

is Tγ = γ + (γ, δ)δ, δ2 = −2

gives T 6= Id,T 2 = Id

HH

base change to make T = Id, X

has node and can resolve it

to X′ → ∆,X′′ → ∆∗

gives 2-families with X′∗ → ∆∗ ∼= X

′′∗ → ∆∗, X ′0∼= X ′′0

but families are different; details also in the appendix tothis lecture.

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I both problems resolved by assuming fibres areI general typeI canonical models (RDP’s are OK)

I What about singular fibres? Normal variety Y hascanonical singularities if for any resolution

Y ′π−→ Y

we have (assuming KY is Cartier, otherwise use Q-linebundles)

KY ′ = π∗KY + E , E = 0

Geometrically, pulling back n-forms from Y we don’t pickup any new poles.The canonical (Weil) divisor class KY on normal surfacesY will also be discussed in the appendix to this lecture.We assume some m0KY is Cartier.

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Example: Assume Y = f (x0, . . . , xn) = 0 in U ⊂ Cn+1

has an isolated singularity at 0,Φ = g(x)dx0 ∧ · · · ∧ dxn/f (x) ∈ Ωn+1

U (Y ) has Poincareresidue

ResY Φ =g(x)dx1 ∧ · · · ∧ dxn

fx0(x)

∣∣∣Y

:= ϕ,

ϕ’s give KY ,0 = ωY ,0.

For hypersurfaces the adjunction conditions are those thatmake the pullback of ResY Φ to any desingularizationY ′ → Y holomorphic.

Suppose blowing up origin Cn+10

π−→ Cn+1 gives aresolution of singularities Y ′

π−→ Y . Then

π∗KY = ResY ′(π∗Φ).

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Exercise: f (x) homogeneous of degree d giving a smoothhypersurface in Pn; then

canonical singularity ⇐⇒ d 5 n

I for the KSBA moduli space of general type curves andsurfaces the conditions are

(a) Xπ−→ ∆ is normal with canonical singularities along X0;

(b) this is true for any base change;(c) ωX/∆ is relatively ample.

Exercise: For curves, (a) + (b) =⇒ X0 has at most nodes.Adding (c) makes all fibres stable.

Definition: For surfaces, (a) + (b) =⇒ X0 has semi-logcanonical (slc) singularities.

Will give the full definition below.

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I If X is normal with an isolated slc singularity a cone overan irreducible plane curve C , then C is either a conic(canonical case), smooth cubic (normal elliptic case) or acuspidal cubic (cusp case).

I If X is a surface with singular locus a double curve Dwith pinch points given locally by zx2 = y 2, then X hasslc singularities.

Locally the normalization (X , D)→ (X ,D) is

(u, v)→ (u, uv , v 2)

the pullback of ResX (dx ∧ dy ∧ dz/zx2 − y 2) isdu ∧ dv/u, so the adjoint conditions imposed on KX by Dare just vanishing on the double curve — don’t see thepinch point in the adjunction conditions.

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We shall usually write (X , D, τ) where τ is the involution

D → D of the double curve branched at the pinch points;τ(0, v) = (0,−v).

I For the resolution f : X ′ → X with f −1(P) =⋃

Ei forisolated normal singularity P

KX ′ ∼Q f ∗KX +∑i

aiEi , ai = −1

is the condition for an isolated slc singularityI Terminal: X smooth, ai > 0;I Canonical: all ai = 0, f ∗KX = KX ′ ;I log-terminal: ai > −1 =⇒ baic = 0;I log-canonical: ai = −1.

I “Semi” comes in essentially by allowing a double curvewith pinch points (more precisely, impose Serre’scondition S2 plus Gorenstein in codimension 1).

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I List of slc singularities (Kollar) is given below.

I Theorem(Horikawa and Kollar–Shepherd-Barron–Alexeev):There exists an irreducible, projective moduli space MH,I

whose general point corresponds to a smooth minimalsurface X of general type and with

K 2X = 2, 1

h2,0(X ) = 2, h1,0(X ) = 0

and whose points of ∂MH,I correspond to minimalsurfaces of general type having slc singularities. Moreover,

dimMH,I = h1(ΘX ) =

26 = h1,1(X )prim−1

28 = h1,1(X )prim.

I Q: What is the structure of ∂MH,I and of the surfacesparametrized by the strata of ∂MH,I?

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I Definition: If X has slc singularities and KX is ample, weshall say that X is stable.

For a stable X with a possibly nodal double curve D, we

shall use models (X , D, τ) of (X ,D) where

I X → X is a resolution of singularities;I D is a reduced normal crossing divisor in X andτ : D → D is an involution that preserves the nodes ofD and where D = D/τ .

I In the list of slc singularities we shall see that the onlyones that can affect the limiting mixed Hodge structureare (3.2.4)(a), (b), (3.3.2), and (3.3.4). Of these those in(3.3.4) are the most intricate and, from aHodge-theoretic point of view, the most interesting.

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I Semi-stable reduction (SSR): A family Xπ−→ ∆ has

normal crossing singularities ifI X is smoothI the Xt = π−1(t) are smooth for t 6= 0; i.e., π is a

submersion on X∗ → ∆∗

I X0 =⋃Xi is a normal crossing variety (NCV)

I locally over the origin π is given in Cn+1 by

x1 · · · xm = t, Xi = xi = 0.

I Theorem: Given any family X′ → ∆′, by base changeand modifications there is non-uniquely a SSR

X //

X′

∆ // ∆′

I One may show that slc singularities have “natural” SSR’s

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Appendix: Some issues in moduliThe purpose of this appendix is to discuss some of the issuesin moduli that are resolved using canonical models of varietiesof general type.

I Jumping of structure: There exists a smooth family ofsmooth surfaces X→ ∆ such that the Xt are allisomorphic for t 6= 0, while Xt is not isomorphic to X0.The standard example is given by taking Et → P1 to begiven by

0→ OP1 → Et → OP1(2)→ 0

with extension class t ∈ H1(OP1(−2)) ∼= C and settingXt = PEt . Then Xt is isomorphic to the Hirzebruchsurface F2 for t 6= 0 while X0

∼= P1 × P1. This jumping ofstructure is due to H0(ΘX0) 6= 0, while for smoothsurfaces X of general type we have H0(ΘX ) = 0. (SeeCatanese for a nice discussion.)

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I Non-separatedness if we require X to be smooth: Thereexist two families X1 → ∆, X2 → ∆ of surfaces that aresmooth, minimal and of general type such that

I X∗1 → ∆∗ is isomorphic to X∗2 → ∆∗ andI X1,0

∼= X2,0,

but where the families Xi → ∆ are not isomorphic. Wemay even assume that the Xi ,t have ample canonicalbundles for t 6= 0.

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The idea of the example is this: Let X′ → ∆′ be a familywhere X′ and the X ′t′ are smooth for t ′ 6= 0 while X ′0 hasan ordinary double point. With the base change t ′ = t2

we obtain X→ ∆ where X has an ordinary double pointP . Resolving the singularity in the standard way replacesP by a quadric Q ∼= P1 × P1. We may then contract X intwo different ways by blowing down P1 × 0 and0 × P1, thus obtaining the Xi → ∆ for i = 1, 2. Theoriginal node on X ′0 is replaced by a −2 curve, so theKXi,0

are nef and big but not ample.

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This problem disappears if we use canonical models ratherthan restricting to smooth, minimal surfaces of generaltype.

I Note: The above problem also cannot occur if the Xi ,0

have ample canonical bundles (cf. Kollar).

I Where does the “−1” in the condition for slc singularitiescome from?We will explain this in the case that X is normal and KX

is a line bundle. The essential geometric idea alreadyoccurs in this case. Let

π : X→ ∆

be a family where X is normal, the fibres over ∆∗ aresmooth and where X denotes the fibre over the origin.

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By semi-stable reduction there is a diagram

X′

π

f // X

π

∆′ // ∆

where ∆′ → ∆ is a base change and X′ is smooth withthe fibre X over the origin being a reduced normalcrossing divisor. Without essential loss of generality wemay also assume that

X = X ′ + Y

where X ′f−→ X is a desingularization and Y ⊂ X meets

X ′ transversely.

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We will also assume for simplicity that ∆′ → ∆ is 1-1, sothat X′ → X is a desingularization.Then we have

KX′ = f ∗KX + E , E = 0,

which gives

KX′ + X ′ = f ∗(KX + X ) + (E − Y ),

and then

KX′ + X ′∣∣∣X ′

= f ∗(

(KX + X )∣∣∣X

)+ (E − Y )

∣∣∣X ′

= =KX ′ = f ∗KX + (E − Y )

∣∣∣X ′.

Since Y meets X ′ transversely we get at most a −1 inpreceding expression for KX ′ .

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Canonical divisor/dualizing sheaf and goodsemi-resolutions (cf. §§5,6 in Kovacs)

I Analytically, for a singular variety X one defines thesections of the canonical sheaf KX to locally be theholomorphic ϕ forms on Xreg such that

´Xreg

ϕ ∧ ϕ is

finite. One also wants a dualizing sheaf ωX , Xreg, and thisrequires more work.

I Let X be a normal surface and ϕ a rational 2-form onXreg with divisor (ϕ) =

∑λiKi and set

KX :=∑

λiK i = Weil divisor

OX (KX ) = f ∈ C(X ) : KX + (f ) = 0 = Weil divisorial sheaf

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I Still assuming that X is normal there is a dualizing sheafωX defined by

ωX = j∗(ωXreg)

where j : Xreg → X is the inclusion, and one has (cf. §5in Kovacs)

ωX = OX (KX ).

I Y is said to be semi-smooth if Ysing is at most a doublecurve DY with pinch points. A proper morphism

Yf−→ X

is a semi-resolution ifI Y is semi-smoothI no component of DY is f -exceptionalI there exists a subvariety Z ⊂ X with codimX Z = 2 such

thatf : f −1(X\Z )

∼−→ X\Z .

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I Let E ⊂ Y be the exceptional divisor. Then f is a goodsemi-resolution if E ⊂ DY is a normal crossing divisor.Good semi-resolutions always (non-uniquely) exist (cf. §6in Kovacs), and it is convenient to work with them. Inlecture 4 we will see a general construction of interestinggood semi-resolutions.

I Note: When the Weil divisor KX must be multiplied tohave a Cartier divisor m0KX so that

f ∗KX :=(

1m0

)f ∗(m0KX ) can be defined we get

denominators in the ai in the expression∑

i aiEi whereai ∈ Q, 0 = a1 = −1. Bounding the denominator in theai is a central issue in the classification problems (cf.Alexeev, Hassett, Laza).

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II. Hodge theoryA. Pure Hodge structures and their variations;period mappings

I Hodge structure of weight n (HS) is (V ,F ) whereI V = Q-vector spaceI F n ⊂ F n−1 ⊂ · · · ⊂ F 0 = VC, satisfies the opposite

condition

F p ⊕ Fn−p+1 ∼−→ VC, 0 5 p 5 n.

I V p,q = F p ∩ Fq, and the opposite condition is

equivalent to the (p, q) decomposition

VC = ⊕p+q=n

V p,q, Vp,q

= V q,p

where F p = ⊕p′=p

V p′,•.

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We setI hp,q = dimV p,q and f p = dimF p

I C = ip−qId on V p,q is the Weil operator

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Let X be a smooth compact complex manifold.

I How does one define F •Hn(X )?

I A•(X ) =n⊕An(X ), d : An(X )→ An+1(X ) with d2 = 0

we have by de Rham

Hn(X ,C) ∼= Hn(A•(X ), d) := HnDR(X ).

I F pA•(X ) = ⊕p′=p

Ap′,• with d(F p) ⊆ F p and

GrpF A•(X ) ∼= ⊕qAp,q(X ).

I The spectral sequence of a filtered complex definesF •Hn(X ).

I Definition: The morphism d of the filtered complexA•(X ) is strict if

F p ∩ dA•(X ) = dF pA•(X )

(definition works for any filtered complex).

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I Theorem: X Kahler =⇒ d is strict and Hn(X ) has aHS of weight n.

I Corollary: X smooth projective variety =⇒ Hn(X ) hasa HS of weight n.

I Corollary: Grp Hn(X ) ∼= ⊕qHp,q

∂(X ) ∼= ⊕

qHp(Ωq

X )

In particular, V n,0 = H0(ΩnX ) = H0(KX ) if dimX = n

I Polarized Hodge structure of weight n (PHS) is (V ,Q,F )where

Q,V ⊗ V → Q, Q(u, v) = (−1)nQ(v , u)

and the Hodge-Riemann (HR) bilinear relations(HRI) : Q(F p,F n−p+1) = 0

(HRII) : Q(u,Cu) > 0, u 6= 0

are satisfied.

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I For n = 2, F n determines the F p’s (F 1 = F 2⊥)I PHS’s form a semi-simple abelian category.

I Example: If λ = c1(L) for L→ X ample, h = (n − d)Idon Hn(X ) where dimX = d , then

[h, λ] = 2λ.

Using Hard Lefschetz

λk : Hd−k(X )∼−→ Hd+k(X )

it follows that there exists a unique sl2 = λ, h, λ+acting on H∗(X ) and preserving the cup product

Q : Hn(X )⊗ H2d−n(X )→ Q

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I sl2-modules decompose into direct sums of

Um := xm, xm−1y , . . . , ym, λ = y∂/∂x , λ+ = x∂/∂y

•

λ •

λ+

ff • • • •

λ •

λ+

ff = N-strings

I this plus Hard-Lefschetz gives for Hd−k(X )prim :=ker λk+1 the primitive decomposition

Hn(X ) = ⊕λkHn−2k(X )prim

and the PHS (Hd−k(X )prim,F ,Q) where

Q(u, v) = Q(λku, v).

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I Example: X = smooth algebraic surfaceH2(X ) = H2(X )prim ⊕Q · λ,H2(X )prim = H2,0(X )⊕ H1,1(X )prim ⊕ H2,0(X ).

I Hodge diamond

rrr

rrr

rrr

rf

fff

f frf’s = primitive stuff

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I Example: Classical description

I´σ ω σ ∈ H1(X ,Z),

σ

I´δ,γ

dxy

0

δ

∞1t

γ

I y2 = x(x − t)(x − 1)

δ

γ

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I period domain D = PHS’s (V ,Q,F ) with given hp,qI compact dual

D = F : Q(F p,F n−p+1) = 0, dimF p = f pI G = Aut(V ,Q) = Q-algebraic group

I GR acts transitively on D = GR/H, H compactI D is a Hermitian symmetric domain ⇐⇒ n = 1 or

n = 2 and h2,0 = 1I GC acts transitively on D = GC/P, P parabolicI D ⊂ D is open GR-orbit

I Usually have VZ ⊂ V and for Γ ⊂ Aut(VZ,Q)

Γ\D =

moduli space for

Γ-equivalence

classes of PHS’s

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I Want to enlarge Γ\D to a Hodge-theoretic completionI Γ\D ⊂ Γ\De , whereI De ⊂ D ∪ ∂D may be described by Lie theory

<>I Canonical identification

I TE Grass(k,VC) ∼= Hom(E ,VC/E )I E means e → [e] where the equivalence class [e] is

defined by: e(t) ∈ Et , e(0) = e and e = de(t)dt

∣∣t=0

I TD ⊂ ⊕Hom(F p,VC/Fp)

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I Definition: Infinitesimal period relation (IPR) is thedistribution I ⊂ TD where

IF = set of F p satisfying F p ⊆ F p−1

I all hp,q 6= 0 =⇒ I is smallest GC-invariant bracketgenerating sub-bundle of TD

I for n = 1 or n = 2 with h2,0 = 1, the IPR is trivial

I for n = 2

• // ((j g c _ [ W T• •

V 2,0 V 1,1 V 0,2

and the IPR is “ //___ = 0 .”

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I for n = 2 and h2,0 = 2, I is a contact system locallygenerated by θ = dy −

∑yidx

i with dθ = −∑

dyi ∧ dx i .Local maximal integrals are 1-jet graphs (Lagrangiansubmanifolds)

(x i)→ (x i , y(x), yx i (x)).

I Definition: A variation of Hodge structure (VHS) isgiven by a locally liftable holomorphic mapping

Φ : S → Γ\D

where lifting Φ : S → D satisfies

Φ∗ : TS → I ⊂ TD.

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I Example: Given Xπ−→ S where X is smooth, projective

and all Xs = π−1(s) are smooth we mayI identify all Hn(Xs)prim with V = Hn(Xs0)prim, up to the

action of the monodromy representation

ρ : π1(S , s0)→ Aut(Hn(Xs0 ,Z)prim)

with image ρ(π1(S , s0)) = Γ.I we then have F p

s ∈ D and the IPR is satisfied.

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I Example: For the I -surface (pg (X ) = 2, q(X ) = 0,K 2

X = 1)Φ∗ : TMI → T (Γ\D)

is injective, and locally the image is a Lagrangiansubmanifold of the contact system.

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B. Mixed Hodge structures and degenerations ofpure Hodge structures

I A mixed Hodge structure (V ,W ,F ) is given byI V = Q-vector spaceI Wk ⊂ · · · ⊂W` = V weight filtrationI F n ⊂ F n−1 ⊂ · · · ⊂ F 0 = VC Hodge filtration

such that on the associated graded

GrWm V = Wm(V )/Wm−1(V )

F p GrWm V := F p ∩Wm + Wm−1/Wm−1

defines a HS of weight m.I MHS’s are iterated extensions of pure HS’s.

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I A degree r morphism of MHS’s isϕ : V → V ′ satisfying

ϕ(Wm) ⊆ W ′m+2r

ϕ(F p) ⊆ F p+r .

It induces a morphism of HS’s

ϕ : GrWm V → GrW′

m+r V′

taking (p, q)→ (p + r , q + r).

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I Morphisms of MHS are strict for both W and F

=⇒ MHS’s form an abelian categoryI Ext1

MHS(A,B) defined, but ExtqMHS(A,B) = 0 for q = 2I Can define a split MHS; can do this over Q or over R.

I Theorem (Deligne): U = complex algebraic variety=⇒ Hn(U) has a canonical MHS.

I U complete =⇒ weight filtration is W0 ⊂ · · · ⊂Wn

I U affine =⇒ weight filtration is Wn ⊂ · · · ⊂W2n.

I Recall that X is a normal crossing variety (NCV) ifX = ∪Xi , Xi smooth and meet transversely

I locally given by x1 · · · xm = 0 in Cn+1

I X [p] :=∐|I |=p

XI , I = (i1, . . . , ip) and XI = ∩Xiα .

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I MHS on Hn(X ) when X = NCVI 0→ CX → ⊕Aq(X [r ]), δ′ = d , δ′′ = RestI E r ,q

1 = Hq(X [r ]), d1 induced by Rest⇒ E r ,q

2 has HS of weight q= d strict for FA•(X [r ])’s =⇒ d2 = d3 = · · · = 0⇒ GrWm Hn(X ) composed of sub-quotients ofHq(X [r ])’s.

I Do MHS’s have (p, q) decompositions?

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I Theorem (Deligne): There exists a unique, functorialdecomposition

VC = ⊕I p,q satisfying

I p,q ≡ Iq,p

modWp+q−2.

I p,q := (F p ∩Wp+q) ∩(F q ∩Wp+q + F q−1 ∩Wp+q−2 + · · ·

)I MHS is R-split ⇐⇒ I p,q = I q,p

I hp,q0 := dim I p,q = Hodge numbers of MHS

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I MHS with (p, q)-decomposition ⊕gp,q on End(V ) andthese we define ∧p,q := ⊕

p′5pq′5q

gp′,q′

I Theorem (Deligne): There exists a unique δ ∈ ∧−1,−1R

such thatF := e−iδF

is R-splitI Gr(V ,W ,F ) ∼= Gr(V ,W , F ).

I F → F means “take real part of the extension data.”

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I Given a nilpotent endomorphism N ∈ End(V ) thereexists a unique filtration

W−n(N) ⊂ W−n+2(N) ⊂ · · · ⊂ Wn(N)

such thatN : Wm(N)→ Wm−2(N)

Nk : Wk(N)∼−→ W−k(N), k = 0.

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I If we have (V ,Q) and N ∈ EndQ(V ), then W•(N) isself-dual

I By Jacobson-Morosov, given N there exists an sl2-tripleN,Y ,N+. Then W•(N) is defined by the N-stringswhen we decompose V

•N // • ··· •

N // •N // •W−n(N)︸ ︷︷ ︸

W−n+1(N)︸ ︷︷ ︸W−n+2(N)

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I Definition: Given (V ,Q), a limiting mixed Hodgestructure (LMHS) is given by a MHS (V ,W (N),F )where F ∈ D, N ∈ EndQ(V ) and

• exp(zN)F ∈ D for Im z 0

• NF p ⊆ F p−1.

I In general F 6∈ DI (V ,W (N), F ) is a LMHS.

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I The conditions for (V ,W (N),F ) to be polarized, alwaysassumed to hold, are

I Set GrW (N)m = Gr

W (N)m (V ,W (N),Fm) for m = 0 and

define

GrW (N)m,prim = kerNm+1 : Gr

W (N)m → Gr

W (N)−m−2

and on GrW (N)m,prim

Qm(u, v) = Q(u,Nmv)

I Polarization condition is that (GrW (N)m,prim,Qm,Fm) is a

PHS.

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I Hodge diamond associated to LMHS has hp,q0 at (p, q)spot and N : I p,q → I p−1,q−1 drawn in

sssh2,0

0

h1,00

h0,00

h2,10 h2,2

0

sss

sss

fff

fffs

Primitive spaces are ’sfsI symmetric about lines p = q and p + q = nI Every Hodge diamond arises from a LMHS (Kerr-Robles)

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I For T = expN so that exp(z + 1)N = T exp(zN) setting

t = e2πiz , z =

(1

2πi

)log t

we obtain a VHS, called a nilpotent orbit

(∗) Φ : ∆∗ → T Z\D

given by

Φ(t) = exp

(1

2πilog t

)N · F .

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I Theorem (Schmid): Any VHS is strongly approximatedby a nilpotent orbit.

I Write (V ,W (N),Flim) for the LMHS that gives theapproximation; it is polarized

I Flim unique up to Flim → exp(zN)Flim; is uniquelyspecified by

T ∗0∆∼= C.

I Example: Given X→ ∆, then by Landman themonodromy on Hn(Xt0) is quasi-unipotent

(Tm − I )n+1 = 0.

I Flim related to the “naive limit” limIm z→∞ =exp(zN) · Flim ∈ ∂D; more on this below.

Flim

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By SSR we may assume T is unipotent with N = logTand that X→ ∆ has X0 = NCV. We set

limt→0

Hn(Xt) := Hnlim(Xt) = (V ,W (N),Flim)

I Example: y 2 = x(x − t)(x − 1)

I 0

δ

∞1tγ

I Φ(t) =

ˆγ

dx

y

/ ˆδ

dx

y∈ TZ

∖H, T = ( 1 1

0 1 )

=

(1

2πi

)log t + a0 + a1t + · · ·

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I Flim = a0 ∈ CI Flim = Re a0 ∈ ∂HI Note that up to equivalence the LMHS only depends on

Φ : ∆∗ → TZ∖D

In particular it is well-defined for KSBA families.

I There exists a spectral sequence that computes Hnlim(Xt)

(Clemens, Steenbrink)I X0 := X and Ep,q

1 = ⊕Ha(X [b])(−c) where (a, b, c)’sdepend on p, q

I d1 = Rest⊕GyI N = “Id” : Ha(X [b])(−c − 1)→ Ha(X [b])(−c) when

both are 6= 0I Ep,q

2 = Ep,q∞ = Gr

W (N)q .

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I Given just a NCV X may define (E1, d1) : d21 = 0 is

implied by Friedman’s condition

OD(X ) ∼= OD

where D = Xsing and OD(X ) is the infinitesimal normalbundle.

I Clemens-Schmid is a long exact sequence of MHS’sderived from the above and where one piece is

Hn(X)→ Hn(Xt)N−→ Hn(Xt)

∼ =

Hn(X )

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I Corollary (local invariant cycle theorem (LICT)):

KerN ∼= ImHn(X )r∗−→ Hn(Xt)

where Xtr−→ X is the degeneration map.

I Recall that KerN = ends of N-strings in Hodgediamonds

rrr

rrr

rrr

rffff

f fI KerN is isomorphic to powers of N applied to primitive

stuff.

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I What about the unique KSBA family X→ ∆?I Will see that LICT remain true on Gr for the I p,0-terms

in the MHS’s (related to Shah — also related to resultthat slc singularities are Du Bois (Kollar-Kovacs))

I In Hodge diamond these are terms on q-axis.

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III. Hodge theory (continued); H and I -surfaces

A. Moduli space of Γ-equivalence classes of PHS’sand its extension

I Reduced limit period mapping takes LMHS’s → LMHS’sand is defined by taking the naive limit

(V ,W (N),F )→ (V ,W (N),F∞)

F∞ = limIm z→∞ exp(zN) · F ∈ ∂D

I I p,q = (p, q)-decomposition for R-split (V ,W (N), F )I (V ,W (N),F∞) is R-split MHS with

F p∞ = ⊕

q5n−pI •,q.

(∗)r r rF 2 rrrF 2

∞

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I Following is very approximate (based on work in progresswith Green, Laza, Robles)

I Given a VHS Φ : S → Γ\D where Z := S\S is locally aNCD with unipotent monodromies, the extended periodmapping is

Φe : S → Γ\De

where in first approximation

I De = D∐(

GrW (N)m,prim ’s for all (N,F )’s

)= D

∐(product of period domains)

I Note that Γ acts on De , and we set

Γ\De :=

extension of the moduli space Γ\Dof Γ-equivalence classes of PHS’s

I Construction of Γ\De depends on the givenΦ : S → Γ\D; cannot be done universally (non-classicalphenomenon)

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I Even as a set the above approximate description of De istoo simple minded; in the non-classical case the IPR willnecessitate additional perhaps discrete data to beincluded.

I Boundary components of De are indexed by Hodgediamonds

I There is concept of Hodge-theoretic incidence denoted−→, between boundary components and criteria for this(Kerr-Robles)

I For n = 2, h2,0 = 2 and h1,1 = r the possible Hodgediamonds are

r r r2

r

2

0 r rr r rr r1

r−21

I r rr r1

r−2

1

II

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r r rrr

2

r−4

III

rr rrr rr

1 1IV rrr r−4

2

V

II

$$III

II

0 // I

<<yyyyyy

""EEE

EE IV // V

III

::uuuuu

I partial but not linear ordering (non-classical)I in general may not even be partial ordering (CY3-folds)

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I How to compute

Φe : ∂S → ∂(Γ\D)

in the geometric case?I GrW (N)(V ,W (N),Flim) = GrW (N)(V ,W (N),F∞)

where F∞ can be read off from the expansion of Φ(t) interms of powers of log t

I For curves and surfaces where the PHS is determined byF n (n = 1, 2), we take ω(t) ∈ H0(Ωn

Xt) and when n = 2

analyze the limit in terms of powers of log tlimω(t) = ω(0) in I 2,0, I 1,0, I 0,0 for the MHS on H2(X0)

I We shall see that this process also works for KSBAlimits.

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H and I -surfaces

I Definition: An H , I -surface is an irreducible, minimalsurface of general type that satisfies

I h2,0(X ) = 2, h1,0(X ) = 0 (Hodge-theoreticassumption)

I K 2X = 2, 1 for H, I surfaces respectively (numerical

assumption)I As discussed in lecture on moduli should allow X to have

canonical singularities.

I Why choose these surfaces?I have small numerical invariantsI close or equal to extremal in terms of Noether’s

inequality

pg (X ) 5

(1

2

)K 2X + 2 (pg (X ) = h2,0(X ))

I will turn out to be described globally by one equation

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I MH,I = KSBA moduli spaceI classically and by Horikawa know that MH,I irreducible

and has dimension h1(ΘX ) =

26

28.

I Concerning the period mapping

Φ : MH,I → Γ\D

Theorem: (i) Φ∗ is injective for I -surfaces andgenerically injective for H-surfaces, (ii) the monodromygroup Γ ⊂ Aut(H2(X ,Z)prim,Q) is arithmetic.

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I In these cases Hodge theory seems a good tool to studymoduli

I for H-surfaces, Φ∗ everywhere injective?I Γ = Aut(H2(X ,Z)prim,Q)?I global Torelli? (Only known non-classical example seems

to be mirror quintic).

I Will focus on H-surfaces:I Hodge theory interacts with algebraic geometry through

H0(Ω2X ) = H0(KX )

where LHS interpreted as subspace of H2(X ,C)(periods) and RHS as canonical series |KX |(algebro-geometric)

I |KX | = pencil whose general member is a smooth curveC with g(C ) = 1

2 (KX · C + C 2) + 1 = 3

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I for general X canonical pencil has two distinct basepoints — blow up gives

X

P1

Ct Cti

E1

E2

, KX

∣∣∣Ct

= ωCtfor all t

tit

I 0→ (m − 1)KX → mKX → Km/2C → 0 and

h1((m − 1)KX

)= 0 for m = 1

I Pm := h0(mKX ) = m(m − 1) + 3, m = 2.

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I H0(KX ) has basis t0, t1 and then H0(2KX ) has basist2

0 , t0t1, t21 , x3, x4 and bicanonical map is

ϕ2KX: X → Q0 ⊂ P4

where Q0 = x0x2 = x21

I for general X , C non-hyperelliptic and ϕ2KXbirational

I ϕ2KX: X → P(1, 1, 2, 2) leading to basic diagram

Xg //

PEπ

f // Q0 ⊂ P4

P1 P1

I E = OP1 ⊕ OP1 ⊕ OP1(2)I PicPE ∼= Zξ ⊕ Zh, ξ = OPE (1) and h = π∗OP1(1)

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I x ∈ |ξ − 2h| generates ξ ⊗ h−2 with divisor(x) = S = P1 × P1

I f = |ξ| is desingularization of Q0, f (S) = Q0,sing

I f g = ϕ2KX“=”ϕ2KX

I g(X ) = X [ has equation

t20xG = F 2, G ∈ |3ξ|,F ∈ |2ξ|

I |KX | has exactly one hyperelliptic curve C0 = t0 = 0I C0 = X [

sing is double curve with 6 + 2 pinch points

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I Picture of X

L

P2t0

P2ti

L = Q0,sing

P2t

∗’s = F = G = 0•’s = basepoints

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I Applications of the picture: |KX | gives VPHS(V,Fp,Q,∇)

I V = R1πZ⊗ OP1

I F1 = π∗ωX/P1∼= OP1(1)⊕ OP1(1)⊕ OP1(3) = E (1)

I VHS determines

PHS on H2(X ,Z)prim

PP X (Torelli type result)

I X has birational model X [ given by one equation

R := L2G − F 2 = 0, R ∈ |4ξ|

Issues are

(i) ξ is big, nef and base-point-free but not ample —contracts S to Q0,sing

(ii) R is not a general member of |4ξ| — has base locusalong which all X [’s are singular.

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I TXMH = ?I Idea is to show that

TXMH∼= TX [Defes(X [)

I For the RHS differentiate the equation

R = (2LG )L + L2G − 2F F ∈ |ξ4 ⊗ J|

where J = Jacobian idealI For zx2 − y2 = 0, J = x2, zx , y; not a regular

sequence but has a manageable syzygy.

I Theorem: TXMH∼= H1(ΘX ) ∼= TX [

(H0(PE ,ξ4⊗J)

Aut(PE)

)I h1(ΘX ) = 26.

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I Generic local TorelliI IF ∼= Hom(V 2,0,V 1,1), so need

TXMH → Hom(H2,0(X ),H1,1(X ))

H1(ΘX )⊗ H0(Ω2X )→ H1(Ω1

X )

I adjunction conditions give KX [∼= ξ ⊗ h−1

I suggests differential Φ∗ is

H0(ξ ⊗ h−1)⊗ H0(ξ4 ⊗ J)

Aut(PE )→ H0(ξ5 ⊗ h−1 ⊗ J)

Aut(PE )

I here cohomology has been lifted from X [ to PE

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I pinch points do not impose adjoint conditions onH0(Ω2

X ) but do impose them on H1(Ω1X )

I unable to show the map has no left kernel for anysmooth X — check for Fermat H-surface

xt20 (x3(t6

0 + t61 ) + r3

0 + r31 ) = (r2

0 + r21 )3

where xt20 , xt0t1, xt

21 , r0, r1 = basis for H0(ξ)

I computation gives h1(Ω1X )prim = 55.

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I IPR for weight n = 2: In TF D have

I V 2,0 //

A

%%V 1,1 // V 0,2 = V 2,0∗

I A ∈ Homalt(V 2,0,V 0,2) ∼= ∧2V 2,0∗

I A = 0 is IPRI contact system for h2,0 = 2I dimD = 2h1,1 + 1; maximal integral manifolds have

dimension h1,1

I for general H-surfaces Φ∗(TXMH) codimension 1 inmaximal integrals

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I For smooth I -surfaces (K 2X = 1, pg (X ) = 2, q(X ) = 0)

I Φ∗ injective everywhereI h1,1 = 57, h1(ΘX ) = 28

=⇒ Φ∗(TXMI ) = Lagrangian submanifoldI locally dy −

∑i y

idxi and integral is

(xi )→ (x , f (x), fxj (x)) = (j1f )(x).

I Theorem: Global monodromy group Γ is an arithmeticgroup.

I Results of this type date to Lefschetz who studied Γ forthe family of smooth sections

Zt = PN−1t ∩Y , Y ⊂ PN smooth of dimension n+ 1

I Regular points Yreg of dual variety correspond to Zt ’swith a node

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I General line S in PN gives a Lefschetz pencil |Zt | withnodal members Zti corresponding to S ∩ Yreg

I For each ti there is a vanishing cycle δi andcorresponding Picard-Lefschetz transformation

Tγ = γ + (−1)εn(γ, δi )δi

I (a) Varying the Lefschetz pencil, the irreducibility ofYreg implies that for PN∗ = PN\Y

the vanishing cycles are conjugate underthe action of π1(PN∗) on Hn(Xs0).

I (b) A topological argument shows that

the vanishing cycles generate the kernel of

Hn(Xt0)→ Hn(Y )

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I (c) Using (a), (b) a group-theoretic argument (cf.Ebeling and Beauville) shows that Γ is of finite index inAut(Hn(Xt0 ,Z)).

I To use this type of argument for H-surfaces we have toextend (a), (b) to a situation where Y = Q0 is singular

I the Zt =ϕ2KXt(Xt) = Q0 ∩ Vt , Vt ∈ |OP4(4)|

I Q0,reg is still irreducible, but for (a) need to analyzeQ0,sing

I Similarly for (b) — use the “picture” of X ’sI (c) requires constructing a degenerate H-surface with a

Q10 singularity.

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I Heuristic principlesI Γ is semi-simple (Deligne-Schmid)I failure of Lefschetz arguments caused by cycles that

intersect singular locus of X ′t ’sI this gives a Γ-invariant sub-Hodge structure in

Hg1(Xt)primI Noether-Lefschetz argument rules this out.

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IV. Period mapping at the boundary

I We begin with the questionWhich slc singularities can contribute to a non-trivialLMHS?

I The oppositie question is: Which slc singularities givefinite monodromy (including T = I )?

I In this case limt→∞H2(Xt) remains in Γ\DI Analogue of compact curvesI Very interesting question that we will not discuss here.

I Motivating the first question is that we want to determinewhich components of ∂MH,I can map to a givencomponent of ∂(Γ\De)

I from the classification of the strata of ∂(Γ\De), answerto the question will indicate where to look.

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I Xπ−→ ∆ is a KSBA family of surfaces of general type with

Xt smooth for t 6= 0 and where X0 = X has slcsingularities.

I By base change may assume monodromy T is unipotentI For X any surface and X = resolution of singularities

(can be reducible), we have non-trivial LMHS if, andonly if,

pg (X ) < pg (Xt).

I Theorem: In the list of slc singularities the only onesthat can contribute to a non-trivial LMHS are

I simple elliptic (3.2.4)(a)

isolatedI cusp (3.2.4)(b)

I cyclic quotient, two branches of D (3.3.2)

nontrivialdouble curveI reducible cases (3.3.4)

In each of these cases the potential contribution to theLMHS can be determined geometrically.

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I Warm up example: For a family of curves acquiring anode

(i) -

P

(ii) -

P

the ω(t) ∈ H0(Ω1Xt

) will specialize to ω(0) ∈ H0(ωX0)

I gives ω(0) on X0π−→ X0, log poles with opposite residues

on π−1(P)

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I ResP ω(0) 6= 0 =⇒ ω(0) 6∈ F 1lim ∩W1(N) and gives

non-zero element in I 1,1

rr rr ResP ω(0)

I ResP ω(0) = 0 =⇒ ω(0) ∈ F 1lim ∩W1(N) and gives

element in I 1,0

I We use the notation

H1,0 → I 0

meaning that Nω(0) ∈ I 0 and we say that “a node maycontribute to the LMHS”

I For global reasons the node actually contributes in case(i) but does not contribute in case (ii).

91/131

I The analogue of this discussion for the slc singularitieslisted above is:

I for the simple elliptic singularity

H2,0 → I 1,0

I for the cuspH2,0 → I 0

I for the cyclic quotient

H2,0 → I 1,0

I for the reducible cases

I 1,0

H2,0

66mmmmmm

((QQQQQ

Q

I 0

both can happen

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I Rules are these:I ω(t)→ ω(0) where ω(0) has log poles on DI Res

Dω(0) is a 1-form that has log poles with opposite

residue lying over the nodes on D; (logarithmic 1-formson D); call the residues at the nodes double residues

I double residues 6= 0 =⇒ ω(0) ∈ I 2,2 and using N2 wewrite

H2,0 → I 0

I double residues = 0 but ResDω(0) 6= 0 gives

H2,0 → I 1,0

I τ : D → D preserves the inverse image ∆ of the nodes(two nodes on D over each node on D) and is branchedover the pinch points

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I for desingularizations D ′ → D we have

D ′/τ → D/τ = D,

and we can read off the I p,0’s for the LMHS from theMHS on X = X0 by

I 2,0 ←→ H0(KX ) = H0(Ω2X

)

I 1,0 ←→ H0(Ω1D′

)−

I 0 ←→ ImH0(Ω1D

(∆))− → H0(O∆)−

I Summary: H0(KX ) are among the 2-forms on the

desingularization X → X thatI have log poles on D whose residues on D are in the −1

eigenspace for the action of τI on D have further log poles lying over the nodes on D

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I since dimH0(KXt ) is constant, we can followω(t) ∈ H0(KXt ) in the limit to determine the I p,0 partsof the LMHS

I these can then be read off as the MHS on H2(X ) by theabove procedure

I since the components of ∂(Γ\De) are indexed bypossible LMHS’s, the above

(i) tells us where to look for degenerations of H andI -surfaces that map to a given Hodge-theoreticboundary component,

(ii) gives a start on the question of classifying whichcomponents of ∂MH,I map to a given component in∂(Γ\De)

I will concentrate on principal components := componentin ∂MH,I where a global singular surface is irreducible.

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I Idea behind the proof of the theorem:I Using the notation from the 1st lecture

f ∗KX ∼Q KX ′ −∑i

aiEi 0 = ai = −1

which leads to

H0(KX ) = H0

(KX ′ +

∑i

b−aicEi

)b−aic =

0 ai 6= −1

1 ai = −1

I Since 2-forms cannot have fractional order poles alongthe Ei we obtain

H0(KX ) → H0

(KX ′ +

∑i

b−aicEi

)→ H0

(KX ′ +

∑i

Ei

)

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I As a first case

H0

(KX ′ +

∑i

Ei

∣∣∣⋃Ei

)= 0 =⇒ H0(KX ′) ∼= H0(KX )

and the LMHS has N = 0 and h2,00 = h2,0

I In general H0

(KX ′ +

∑i Ei

∣∣∣⋃Ei

)= H0

(KX+D

∣∣∣D

)=

logarithmic1-forms on D

where K

X+D:= 2-forms with log-poles on D.

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I If there is one Ei meeting⋃

j 6=i Ej in one point, then wemay erase Ei without affecting the logarithmic 1-formson D. Thus we may eliminate all trees in Kollar’s list,and then by inspection we obtain the theorem.

I Seems plausible that one may in fact determine

GrW (N)2 (LMHS)tr

from X in a KSBA degeneration.

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Realization of the Hodge-theoretic boundary strataby the boundary of the KSBA moduli space for Hand I -surfaces

I Stratification of Γ\De is

II&&LL

LLL

0 // I

::uuuuu

$$IIII IV // V

III

88rrrr

Want to show that the extended period mapping

Φe : ∂MH,I → ∂(Γ\D)

realizes the Hodge-theoretic stratification in the sensethat

I there exist points in every stratum in ∂(Γ\D) that are inthe image of Φe

I there exist degenerations between strata in ∂MH,I thatmap onto Hodge-theoretic degenerations between stratain ∂(Γ\D).

99/131

I This is only a first step towards understanding ∂MH,I

using Hodge theory

I Q: Is there a surface of general type with dimM > 0 forwhich the above result fails?

I We shall prove the above result for H-surfaces and fordegenerations 0→ I of smooth surfaces to ones whoseLMHS is

r rr r rr r1 1

100/131

I Idea isI First do a thought exampleI Then using this as a guide, from the equation of H

surfaces write down an actual exampleI Other examples of realizing Hodge-theoretic

degenerations are given in the appendix to this lecture.

I Want to construct a degeneration Xt → X under whichI one holomorphic 2-form ω(t) remains holomorphic in the

limit; ω(0) ∈ H0(Ω2X

)I a general holomorphic 2-form ϕ(t) becomes singular and

ResD ϕ(0) contributes to I 1,0

I to construct ω(t) suppose that the divisor Dt = (ω(t))specializes to the double curve D in the limit; locallysomething like

101/131

ω(t) = ResXt

(f (x , y , z)dx ∧ dy ∧ dz

xy − t

)↓

ω(0) = ResX

(f (x , y , z)dx ∧ dy ∧ dz

xy

)=

f (x , 0, z)dx ∧ dz

x

∣∣∣y=0− f (0, y , z)dy ∧ dz

y

∣∣∣x=0

which will not have a non-zero log-pole onD = x = 0, y = 0 when

f (x , y , z) = xg(x , y , z) + yh(x , y , z)

and then ω(0) ∈ H0(Ω2X

).

102/131

I On the other hand, for ϕ(t) = ResXt

(f (x ,y ,z)dx∧dy∧dz)xy−t if

f (x , y , z) does not vanish on D then

ResD ResX (ϕ(0)) ∈ H0(Ω1D

)−

contributes to I 1,0.

I Recalling the picture

X [t =

××××××

rt0=0

, xt20Gt = F 2

t

where |K [Xt| = fibres and t0 = 0 is the double curve with

pinch points on X [t suggests considering

X [ = xt20xt

21 · Q = F 2

to realize the thought example

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I the fibre over t1 = 0 on X [t has become a new double

curve on X [0 .

I Proposition: The normalizations of the X [t give a family

X∗ → ∆∗ of H-surfaces with KSBA completion Xπ−→ ∆.

The extended period map Φe : ∆→ Γ\De maps theorigin to a type I degeneration and the normalization ofπ−1(0) is a K3 surface.

The sketch of the proof together with some furtherdegenerations will be given in the appendix to this lecture.

104/131

Extension of the Liu-Rollenske construction andmaximally degenerate I -surfaces

I Type V Hodge-theoretic boundary consists of theHodge-Tate Gr(LMHS)’s

rrr r 2

I Degenerate curves and surfaces where LMHS isHodge-Tate may have moduli — for curves an example is

105/131

I Normalization is P1 with four marked points whosecross-ratio gives the moduli — we shall see surfaceanalogues of this

I Of particular interest are maximally degenerate curvesand surfaces X , meaning that the LMHS is Hodge-Tateand that X has no equisingular deformations; i.e.

TXDefes(X ) = H1(Ext1OX

(Ω1X ,OX )) = 0

I For curves the rigid ones are configurations of(P1, 0, 1,∞)’s; i.e. stable curves whose dual graph istrivalent

106/131

I For surfaces one may hope that interesting rigid onesmay be constructed from configurations of(P2, L1, L2, L3, L4)’s, where the Li are lines in generalposition.

I Theorem: There exists a (unique?) rigid I -surface Xwhose LMHS is of type V. Its normalization is(X , D, τ) = (P2, L1 + L2 + L3 + L4, τ), where τ will bedescribed below.

I Recall that a smooth I -surface X is minimal, general typewith

pg (X ) = 2, q(X ) = 0

K 2X = 1

and 2:1 bicanonical map

ϕ2KX: X → Q0 ⊂ P3,

Q0 = x0x2 = x21 =

P

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I branch locus is P + V ∩ Q0, V ∈ |OP3(5)| general: P =base point.

I |KX | = 2:1 branched covering of Q0.

I Given Y = smooth surface and C1,C2 = disjoint nodalcurves with identification

f : C1∼−→ C2

may construct Y /f = surface of type (3.3.4) singularityfrom Kollar’s list

I Will consider E ⊂ Y = smooth curve meeting C1,C2

with f (E ∩ C1) = E ∩ C2, and

ω ∈ H0(Ω2X (C1 + C2 + E )

)ResC1 ω = −f ∗ResC2 ω, and

double residues at C1 ∩ E are minus those at C2 ∩ E .

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I Following Liu-Rollenske start with

L4

L3

L2 L1

in P2

I want to construct quotient surface by identifyingL3 ←→ L4, L1 ←→ L2; two aspects:

(i) lines not disjoint so need to blow up at Pij = Li ∩ Lj toEij

(ii) identification (P1; 3 points)∼−→ (P1, 3 points) needs to

specify which 3 points∼−→ which 3 points

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3 32

L1 L2

L3

L43

2 1

21

4

14

4

I “2” on L1 means P12, line 2 r r1 means E12 = E21, etc.

L3 ←→ L4 L1←→ L2

1←→ 2 2 ←→ 1

2←→ 1 3 ←→ 3

4←→ 3 4 ←→ 4

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I Identifying L3 ←→ L4 gives

L1 L2

α α

β β

δ

γ γ

I identifying L1 ←→ L2

D12

αD34

δ γα, β, γ, δ = singular points

β

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I α, β, γ, δ are isolated singularities withI α, δ = cone over nodal cubicI β, γ = x2 + y2z2 = 0 (= T2,∞,∞ singularity)I |KX | = pencil of lines through P

L4

P

L3

L2 L1

K 2X = 1

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I bicanonical image branched over

P

@@I

︸ ︷︷ ︸P + [P2

0 + 2(︷ ︸︸ ︷P2

1 + P22)] ∩ Q0

BBBBM

I equation in P(1, 1, 2, 5) is

z2 = y(x1 − y)2(x2 − y)2

I bicanonical image has no non-trivial equisingulardeformations

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Appendix

Let

Xs =

surface given by the normalization

of the surface xt20 (sG − xt2

1Q) = F 2

and then contracting the −1 curves.

Here, G and F ,Q are general elements of |3ξ|, |2ξ|.

I Proposition: The above gives a smooth family X∗ → ∆∗

of H-surfaces. The extended period map Φe : ∆→ Γ\De

maps the origin to a type I degeneration. Thenormalization of X0 is a K3 surface.

Also, X→ ∆ is a KSBA degeneration.

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I Sketch of Proof:I x2t2

0 t21Q = F 2 is the equation of X [

0 ⊂ PEI X [

0 has double curve

(x = 0 ∪ t0 = 0 ∪ t1 = 0) ∩ F = 0

I the divisor S = (x) of x ∈ |ξ − 2h| is P1 × P1 whichunder |ξ| : PE → P4 maps to the singular line on

Q0 = x0x2 = x21 ⊂ P4.

I Aside: It is easier to see this picture one dimension down— take F = OP1 ⊕ OP1(2) to get PF → P1. Then

H0(PF , ξ) has basis t20 , t0t1, t

22 , x3

and the map PF → P3 is

x0 = t20 , x1 = t0t2, x2 = t2

2 , x3 = x3

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and this gives a mapPF→Q′0⊂P3

=

x0x2=x21

H0(ξ − 2h) = Cx where (x) = P1

(x) -

P

The divisor (x) = P1 with (x)2 = −2 contracts to thedouble point P = Q ′0.The map PE → Q0 ⊂ P4 is like this with P ↔ singularline of Q0 and (x) ∼= P1 × P1 maps to Q0,sing.

<>

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I F∣∣S∈ |OP1×P1(0, 2)|; thus

I S ∩ X [ is a P1 q P1 corresponding to the images ofE1,E2 in X = blow up of X at base points of |KX |

I S ∩ X [0 is 2(P1 q P1) corresponding to the x2 in the

equation of X [0 . This double curve has no pinch points.

I The other double curves on X [0 are on t0 = 0 and t1 = 0

with pinch points given respectively by

t0 = 0 ∩ Q = F = 0t1 = 0 ∩ Q = F = 0.

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More precisely,I on x 6= 0, t0 = 0 they are given by F = Q = 0 on

P2t0\P1

=

P2t0∩ x = 0

which is four pointsI on x = 0 and t0 = 0 the equation is locally of the form

u2v2 − w2 = 0

=(uv − w)(uv + w)

which is a double curve without pinch points — similarthing happens on t2 = 0

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I the normalization X [0 of X [

0 has the picture

Ct0 Ct1

E22

E21

E12

E11

t0=0 t1=0

where Ct0 ,Ct1 are elliptic curves branched over P1 at thefour pinch points. The Eij are the P1’s that arise from thenormalization of the 2(P1 q P1) above. The involution ofCt0 → P1 interchanges E11 ∩ Ct0 and E12 ∩ Ct0 , etc.

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I Issue is that X [0 is not the normalization X0 of the KSBA

limit X → X0. The reason is that the double curve t0 = 0on X [ is a singularity of ϕ2KX

(X ) and not a singularity ofX . The correct interpretation is that

The limit of Ct0 as X → X0 is a P1 on X0, while the

limit of Ct1 on X0 is an elliptic curve with Ct1 → P1

branched at the four pinch points.

I The normalization = desingularization X0 of X0 is a K3surface with ω0 ∈ H0(KX0

) non-vanishing.

I Reason: The divisor (ω) on X is 2(E1 + E2) + Ct1 . On Xthe E1,E2 are contracted. Moreover as X → X0, thedivisor Ct1 of ω tends to a double curve on X0 along which

ω0 is regular and 6= 0. Thus on X0, ω0 gives KX0= OX0

.

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I Note: The construction may be reversed as follows:Start with a K3 having two elliptic curves C0,C1 meetingin four points.

Blow up the four points to obtain

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I Assume now that each of C0,C1 has an involution, andthat C0 ∩ C1 are two pairs where the two points in eachpair are interchanged by the involution. Then glue thetwo pairs of horizontal P’s together to obtain a surfacewith two double curves

The singular curves are

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Now contract C0 to a P1 using the involution. This is alittle subtle as because of the x3 in the equation thesingular curves over t1 = 0, t0 + t1 = 0 will have anequation

y 2 = (x − a1)3(x − a2)3(x − a3)(x − a4).

Using the birational transformationu = x

v = y/(x − a1)(x − a2)

on P2, with inversex = u

y = v(u − a1)(u − a2)

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the above equation becomes one of the form

u2 = (v − b1)(v − b2)(v − b3)(v − b4).

Going through an analysis as before we find on X0 twoelliptic curves that give rise to the I 1,0 in the LMHS.

I For a type II degeneration with Hodge diamond

r rr r1 1

dim I 2,0 = 1

dim I 1,0 = 1

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what is suggested is that we take a type I degenerationwith Hodge diamond

r rr r rr r1 1

dim I 2,0 = 1

dim I 1,0 = 1

and degenerate the I 1,0 to I 0,0. For the equation of a typeI degeneration

xt20xt

21Q = F

where Q,F are general, what is then suggested is that welet the conics Q = 0, F = 0 in the P2

t1given by t1 = 0

become special;

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e.g., if we take them to be

F=0Q=0

then if the curve Ct1 for a type I degeneration has theequation

y 2 = (x − a1)(x − a2)(x − a3)(x − a4)

the degeneration equation is

y 2 = (x − b1)2(x − b2)2

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which is a reducible curve consisting of a pair of P1’smeeting in 2-points

in P2

The holomorphic 2-form with a log-pole on ct1 in the typeI degeneration then becomes one whose residue is theunique 1-form on the above curve with log-poles at thenodes — This is the I 0,0.

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Selected References

I D. Abramovich and K. Karu, Weak semistable reductionin characteristic 0, Invent. Math. 139 (2000), 241–273.

I V. Alexeev, Boundedness and K 2 for log surfaces,Internat. J. Math. 5 (1994), no. 6, 779–810.

I W. Barth, K. Hulek, C. Peters and A. van de Ven,Compact Complex Surfaces, Ergeb. Math. Grenzgeb. 4,436pp., Springer-Verlag, New York, 2004.

I F. Catanese, A superficial working guide to deformationsand moduli, in Handbook of Moduli, Vol. I, Adv.Lectures in Math. 25, pp. 161–216, (G. Farkas and I.Morrison, eds.), Int. Press, Somerville, Ma, 2013..

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I E. Cattani and A. Kaplan, Polarized mixed Hodgesstructures and the local monodromy of a variation ofHodge structure, Invent. Math. 67 (1982), no. 1,101–115.

I E. Cattani, A. Kaplan, and W. Schmid, Degeneration ofHodge structures, Ann. of Math. 123 (1986), no. 3,457–535.

I W. Ebeling, An arithmetic characterisation of thesymmetric monodromy group of singularities, Invent.Math. 77 (1984), 85–99.

I R. Friedman, Global smoothings of varieties with normalcrossings, Ann. of Math. 118 (1983), 75–114.

I B. Hassett, Moduli spaces of weighted pointed stablecurves, Adv. Math. 173 (2003), no. 2., 316–352.

I E. Horikawa, Algebraic surfaces of general type with smallc2

1 . IV, Invent. Math. 50 (1979), 103–128.

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I M. Kerr and C. Robles, Partial orders and polarizedrelations on limit mixed Hodge structures, in preparation,2015.

I J. Kollar, Moduli of varieties of general type, inHandbook of Moduli, Vol. II, Adv. Lectures in Math. 25,pp. 131–158, (G. Farkas and I. Morrison, eds.), Int.Press, Somerville, Ma, 2013..

I J. Kollar and S. J. Kovacs, Log canonical singularities areDu Bois, J. Amer. Math. Soc. 23 (2010), no. 3,791–813.

I J. Kollar and N. I. Shepherd-Barron, Threefolds anddeformations of surface singularities, Invent. Math. 91(1988), 299–338.

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I S. J. Kovacs, Singularities of stable varieties, inHandbook of Moduli, Vol. II, Adv. Lectures in Math. 25,pp. 159–204, (G. Farkas and I. Morrison, eds.), Int.Press, Somerville, Ma, 2013..

I A. Landman, On the Picard-Lefschetz transformation foralgebraic manifolds acquiring general singularities, Trans.Amer. Math. Soc. 181 (1973), 89–126.

I R. Laza, The KSBA compactification for the modulispace of degree two K3 pairs, 2014.

I W. Liu and S. Rollenske, Geography of Gorenstein stablelog surfaces, Trans. Amer. Math. Soc. 368 (2016), no.4., 2563–2588.

I V. P. Palamodov, Deformation of complex spaces,Russian Math. Surveys 31 (1976), no. 3, 129–197.

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I C. Peters and J. H. M. Steenbrink, Mixed HodgeStructure, Ergeb. Math. Grenzeb. 52, 470pp.,Springer-Verlag, New York, 2004.

I W. Schmid, Variation of Hodge structure: thesingularities of the period mapping, Invent. Math. 22(1973), 211–319.

I J. Shah, Insignificant limit singularities of surfaces andtheir mixed Hodge structure, Ann. of Math. 109 (1979),no. 3, 497–536.