On the geography of symplectic manifolds · Then we recall Darboux’s theorem, which states that any two symplectic forms are locally isomorphic, so that symplectic invariants must

On the geography of symplectic manifolds

Proefschrift

ter verkrijging vande graad van Doctor aan de Universiteit Leiden,

op gezag van de Rector Magnificus Dr. D. D. Breimer,hoogleraar in de faculteit der Wiskunde en

Natuurwetenschappen en die der Geneeskunde,volgens besluit van het College voor Promoties

te verdedigen op donderdag 24 juni 2004klokke 14.15 uur

door

Federica Benedetta Pasquotto

geboren te Verona, Italiein 1974

Samenstelling van de promotiecommissie:

promotor: prof. dr. H. Geigesreferent: prof. dr. C. B. Thomas (University of Cambridge)overige leden: prof. dr. G. van Dijk

prof. dr. S. J. Edixhovendr. M. Lubkeprof. dr. A. I. Stipsicz (A. Renyi Institute, Budapest)

On the geography

of symplectic manifolds.

Federica PasquottoMathematisch Instituut, Universiteit Leiden, The Netherlands

ISBN: 90-9018155-5

Printed by Universal Press

Introduction.

The subject of this thesis is symplectic topology. More specifically, we are interested inconstruction and invariants of symplectic manifolds.

Differential topology and algebraic geometry provide us with some standard operationswhich can be performed within the differentiable and complex category, respectively. For in-stance: connected sum along a submanifold, blow-up, construction of fibrations and branchedcoverings. These operations make sense, under certain conditions, in the symplectic categoryand thus enable us to produce new examples of manifolds admitting symplectic structures.

Given a symplectic manifold (M,ω) of dimension 2n, one can define its Chern classesas the Chern classes of a tame almost complex structure J. In general, the Chern classesof an almost complex manifold, that is, a manifold with a complex structure on the tangentbundle, only depend on a connected choice of such complex structure. Since the space oftame almost complex structures for a given symplectic form is connected, the Chern classesof a symplectic manifold are invariants of the symplectic form.

By evaluating top-dimensional products of Chern classes on the fundamental homologyclass of M, one obtains a system of integer numbers, in fact as many as the partitions of n,which are called the Chern numbers of (M,ω). The problem of determining which combi-nations of integer numbers may appear as Chern numbers of a closed, connected, symplecticmanifold is known in the literature under the name of symplectic geography.

Symplectic geography is a suitable subject of study when considering symplectic andrelated structures, in particular almost complex and Kahler structures. Symplectic manifolds,in fact, occupy the central position in the sequence of inclusions

Kahler ( symplectic ( almost complex.

These inclusions have long been known to be proper.One is interested in finding out which properties distinguish symplectic manifolds from

the manifolds in the other two classes and which ones do not. Chern numbers can be definedfor every (closed) almost complex manifold, so we can compare the geography of manifoldsbelonging to all classes. The main theorem in this thesis states that in dimension 8 the geo-graphy of symplectic manifolds does not differ from that of almost complex ones.

Theorem. Any ordered quintuple of integers which arises as the system of Chern numbersof an almost complex 8-dimensional manifold can also be realised by a closed, connected,symplectic 8-manifold.

i

ii INTRODUCTION.

The analogous result in dimension 6 is due to Halic ([13]). His proof makes use of twoimportant operations: blow-up and connected symplectic sum. Both of them may be per-formed inside the symplectic category. We also consider symplectic fibrations, obtained byprojectifying a complex vector bundle over a symplectic manifold, and symplectic branchedcoverings with a given symplectic basis and branching set.

This thesis consists of five parts or, more precisely, four chapters and one appendix.The first chapter is a collection of some basic definitions and results of symplectic geome-

try. We define here manifolds, submanifolds and isomorphisms in the symplectic category.Then we recall Darboux’s theorem, which states that any two symplectic forms are locallyisomorphic, so that symplectic invariants must necessarily be of a global nature. After this,we focus on the relationship between symplectic and almost complex manifolds on one sideand symplectic and Kahler manifolds on the other. We hope to have included all the notionswhich are necessary to comprehend the following material.

The second chapter deals with several ways of constructing symplectic manifolds. Werecall some standard operations which may be performed on one or more given symplecticmanifolds, to produce a new manifold which again admits a symplectic structure. Someof these constructions come from complex geometry (for instance, blow-up), others are ofpurely topological nature (for instance, connected sum along a symplectic submanifold). Wealso show that the branched covering of a symplectic manifold, branched along a symplecticsubmanifold, must admit a symplectic structure and apply this result to the construction ofcyclic branched coverings with a given branching set.

The third chapter starts dealing with invariants of symplectic manifolds, namely with theirChern classes. In particular, we try to describe how to compute them efficiently for the mani-folds obtained by performing the constructions of Chapter 2. This is possible because for thesymplectic forms arising from these operations there is a well defined, unique homotopy classof almost complex structures. The main section of this chapter is the one about Chern classesof blow-ups. The “blow-up formula” which we obtain was already known for algebraic vari-eties and we show how the proof may be modified in order to apply to symplectic manifolds.Due to their length, the computations of the invariants in dimension 8 are postponed to theappendix. In this chapter we also apply Donaldson’s result about the existence of symplecticsubmanifolds to the total space of some given symplectic fibrations and compute the Chernclasses and numbers of such submanifolds.

The fourth chapter finally studies Chern numbers and the geography of closed, symplec-tic, 8-dimensional manifolds. The first part of it, though, is completely devoted to a reviewof the main facts about the complex cobordism ring of Milnor and its relationship with Chernnumbers of almost complex manifolds. More precisely, we describe how knowledge of thestructure of this ring (it is a polynomial ring and Milnor was able to point out explicit ge-nerators for it) eventually enables one to write down some congruence relations which mustbe satisfied by the Chern numbers of any almost complex manifold, hence any symplecticmanifold as well. In dimension 8, for instance, these relations are

−c4 + c1c3 +3c22 +4c2c2

1 − c41 ≡ 0 (mod 720)

2c41 + c2

1c2 ≡ 0 (mod 12) (1)

−2c4 + c1c3 ≡ 0 (mod 4).

iii

We thought this review worth the effort, since there seems to be an inclination in the literaturetowards referring for these matters to a “mythical” part 2 of the paper [21] of Milnor, whichwas in fact never published. After this overview, we start constructing the examples that willfill our 8-dimensional symplectic geography picture, compute their Chern numbers (the ex-plicit computations are in fact once more relegated to the appendix) and show that, preciselyas in the case of almost complex manifolds, we are able to obtain all combinations of num-bers which are allowed by system (1) as Chern numbers of a closed, connected, symplectic8-dimensional manifold.

In the fifth part, the appendix, we carry out the computations: the blow-up formula isapplied to get expressions for the Chern classes of blow-ups in dimension 8. From this thetop Chern classes and subsequently the Chern numbers are calculated, by using informationon the structure of the cohomology ring of the blow-up. We have also collected here thecomputations of the Chern numbers of the sumbanifolds obtained in Chapter 3 by applyingDonaldson’s theorem.

To conclude this introduction, we would like to point out an interesting potential applica-tion of symplectic geography. We have seen that Chern numbers of a symplectic manifold areinvariants of the symplectic form. When considering invariants of some nature, it is alwaysinteresting and natural to ask to what extent these invariants classify. The Chern numberscertainly fail to classify symplectic structures. The homotopy class of tame almost complexstructures is an invariant of deformation equivalence, so deformation equivalent forms havethe same Chern numbers. Isomorphic symplectic forms also have the same numbers. Thereare even examples of symplectic forms which are not related by any sequence of isomor-phisms and deformation equivalences and which are distinguished by finer invariants (Chernclasses, Gromov-Witten invariants), but not by the Chern numbers. In other words, the extentto which Chern numbers fail to classify symplectic structures is considerable. Therefore onemay wonder whether they might not be topological invariants. In dimension 4, the Chernnumbers c2

1 and c2 are indeed topological invariants. In dimension 6, LeBrun has shownthat Chern numbers are not topological invariants of complex manifolds, but what happensif we introduce a symplectic form is not known. A better understanding of the geographyof symplectic manifolds may be useful in order to answer this question by comparing theChern numbers with the topology. The symplectic constructions on which Halic’s results andour main theorem rely allow a good control of the cohomological data. In dimension 6 and8 there are smooth classification theorems (Wall [30] and Muller [23], respectively) basedon those data. So one might hope to be able to detect a smooth manifold realising two dif-ferent combinations of Chern numbers, that is, admitting two distinct symplectic structures,distinguished by the Chern numbers.

iv INTRODUCTION.

Contents

Introduction. i

1 Basic notions. 11.1 Symplectic manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Submanifolds and tubular neighbourhoods. . . . . . . . . . . . . . . . . . . 21.3 Symplectic and almost complex structures. . . . . . . . . . . . . . . . . . . . 21.4 Kahler manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 4-manifolds and the intersection form. . . . . . . . . . . . . . . . . . . . . . 41.6 Pseudo-holomorphic curves. . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Construction of symplectic manifolds. 72.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Thurston’s construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Symplectic blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Blow-up at a point. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Symplectic form on the blow-up. . . . . . . . . . . . . . . . . . . . . 112.3.3 Blow-up along a submanifold. . . . . . . . . . . . . . . . . . . . . . 11

2.4 Symplectic connected sum. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Branched coverings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5.2 Cyclic branched coverings. . . . . . . . . . . . . . . . . . . . . . . . 152.5.3 Symplectic structures on branched coverings. . . . . . . . . . . . . . 17

3 Chern classes of symplectic manifolds. 213.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Symplectic sphere bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Chern classes of projective bundles. . . . . . . . . . . . . . . . . . . 223.2.2 Cyclic branched coverings. . . . . . . . . . . . . . . . . . . . . . . . 253.2.3 Other submanifolds: sections. . . . . . . . . . . . . . . . . . . . . . 273.2.4 Donaldson’s theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.5 Branched coverings as submanifolds. . . . . . . . . . . . . . . . . . 29

3.3 Chern classes of blow-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.1 Some cohomological lemmas. . . . . . . . . . . . . . . . . . . . . . 30

v

vi CONTENTS

3.3.2 Remark on the definition of Chern classes of blow-up. . . . . . . . . 333.3.3 The blow-up formula. . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Symplectic sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.1 Symplectic sum along surfaces with trivial normal bundle. . . . . . . 373.4.2 Symplectic sums along tori in dimension 4. . . . . . . . . . . . . . . 38

4 Symplectic geography. 414.1 Cobordism ring and Chern numbers. . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Stable equivalence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.2 Hypersurfaces of bidegree (1,1). . . . . . . . . . . . . . . . . . . . . 444.1.3 The complex cobordism ring. . . . . . . . . . . . . . . . . . . . . . 454.1.4 The notion of C-equivalence. . . . . . . . . . . . . . . . . . . . . . . 46

4.2 The geography problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.1 The theorem of Riemann-Roch. . . . . . . . . . . . . . . . . . . . . 504.2.2 Geography of symplectic manifolds. . . . . . . . . . . . . . . . . . . 51

4.3 The eight-dimensional case. . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 Congruence relations in dimension eight. . . . . . . . . . . . . . . . 534.3.2 The symplectic case. . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3.3 Behaviour of the parameters under blow-up. . . . . . . . . . . . . . . 57

4.4 Building blocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4.1 Elliptic surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4.2 Other building blocks. . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5 Construction of the examples. . . . . . . . . . . . . . . . . . . . . . . . . . 614.5.1 Symplectic sphere bundles, Part II. . . . . . . . . . . . . . . . . . . . 61

4.6 The blow-up systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.6.1 Realising sets of parameters with j ≥ 1. . . . . . . . . . . . . . . . . 654.6.2 The case j = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.6.3 Negative values of j. . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.7 Some final remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.7.1 Kahler manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.7.2 Geography with fundamental group. . . . . . . . . . . . . . . . . . . 70

A Some computations. 73A.1 Chern numbers of blow-up in dimension 8. . . . . . . . . . . . . . . . . . . 73

A.1.1 Blow-up at a point. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74A.1.2 Blow-up along a curve. . . . . . . . . . . . . . . . . . . . . . . . . . 76A.1.3 Blow-up along a four-dimensional submanifold. . . . . . . . . . . . 78

A.2 Submanifolds from Donaldson’s theorem. . . . . . . . . . . . . . . . . . . . 81

Samenvatting. 87

Acknowledgements. 89

Curriculum Vitae. 91

Chapter 1

Basic notions.

1.1 Symplectic manifolds.

Definition 1.1. A symplectic manifold is a smooth manifold M with a closed nondegeneratetwo-form ω. Nondegeneracy means that the top exterior power of ω is nowhere zero, i.e.,it is a volume form. Thus, M is always even dimensional and canonically oriented. A mapf : M→N between symplectic manifolds (M,ωM) and (N,ωN) is called symplectic if f ∗ωN =ωM ; a symplectic diffeomorphism is called a symplectomorphism.

Example 1.2. Euclidean space R2n with coordinates (x1, · · · ,xn,y1, · · · ,yn) and the form

ω0 =n

∑i=1

dxi ∧dyi

is a symplectic manifold. The form ω0 is called standard or canonical symplectic form.

In fact, Darboux’s theorem states that the above example is universal, in the followingsense (see [20, p. 95]):

Theorem 1.3. Every symplectic manifold (M,ω) is locally isomorphic to euclidean spacewith the standard symplectic form.

So there are no symplectic local invariants; globally, of course, the situation is different:volume, for example, is preserved by symplectic isomorphisms.

Definition 1.4. Two symplectic forms ω0 and ω1 on M are said to be isotopic if they can bejoined by a smooth family of cohomologous symplectic forms ωt on M, strongly isotopic ifthere is an isotopy Ft of M such that F∗

1 ω1 = ω0.

In general, of course, strong isotopy implies isotopy (by setting ωt = F∗t ω1), but for a

closed manifold the opposite implication is also true as a corollary of the following theorem.

Theorem 1.5 (Moser stability theorem). If ωt is a smooth family of cohomologous sym-plectic forms on a closed manifold M, there exists an isotopy F of the identity of M such thatF∗

t ωt = ω0 for all t.

1

2 CHAPTER 1. BASIC NOTIONS.

1.2 Submanifolds and tubular neighbourhoods.

Definition 1.6. A smooth submanifold N of a symplectic manifold (M,ω) is called a sym-plectic submanifold if ω restricts to a symplectic form on TN.

In this case the normal bundle of N in M may be identified with the symplectic comple-ment of the tangent bundle, namely the bundle T Nω defined pointwise at p ∈ N by

(T Nω)p = v ∈ TpM | ω(v,w) = 0 for all w ∈ TpN.

Then a neighbourhood of N is completely determined by the isomorphism class of the normalbundle. This result is referred to as the Symplectic Neighbourhood Theorem and can be foundin [20, p. 101], where it is stated more precisely in the form below.

Proposition 1.7. Let (Mi,ωi), i = 1,2, be symplectic manifolds. Suppose we are given sym-plectic embeddings j1 and j2 of N in M1 and M2, respectively, such that the normal bundlesof the two embeddings are isomorphic. Then there exists tubular neighbourhoods Ui of ji(N)and a symplectomorphism φ : U1→U2 such that the differential of φ induces between thenormal bundles the given isomorphism.

In particular, if N is a symplectic submanifold of M, a tubular neighbourhood of N isalways symplectomorphic to a tubular neighbourhood of the zero section of the normal bundleof N in M.

1.3 Symplectic and almost complex structures.

Definition 1.8. An almost complex structure on a smooth oriented manifold M is an iso-morphism of the tangent bundle J : T M→T M such that J2 = −idT M . In other words, analmost complex structure on M is a complex structure on its tangent bundle. Thus M ad-mits an almost complex structure if and only if the structure group of T M may be reducedfrom SO(2n) to the unitary group U(n). A nondegenerate 2-form ω ∈ Ω2(M) and an almostcomplex structure J on M are called compatible if the bilinear form

〈v,w〉 := ω(v,Jw)

defines a Riemannian metric on M.

Notice that ω and J are compatible if and only if the following two conditions are satisfied:

(i) ω(Jv,Jw) = ω(v,w) for all v,w ∈ TpM and p ∈ M;

(ii) ω(v,Jv) > 0 for all v ∈ TpM, v 6= 0, and p ∈ M .

If (ii) alone holds, one says that ω tames J or that J is a tame almost complex structure. Thetaming conditon alone already implies that ω is nondegenerate, hence a closed taming 2-formis automatically symplectic.

If we fix a nondegenerate 2-form, for example a symplectic form, there isn’t a uniquetame almost complex structure. However, we have the following result (for a proof, see [20,p. 118]).

1.4. KAHLER MANIFOLDS. 3

Proposition 1.9. Given a nondegenerate 2-form ω, the corresponding spaces of compatibleand tame almost complex structures are nonempty and contractible.

Given a complex vector bundle (ξ,J) with base M, one can define its Chern classesci(ξ) ∈ H i(M;Z). If the complex rank of ξ is r, one denotes by c(ξ) the total Chern class∑r

i=0 ci(ξ). These classes depend on a connected choice of complex structure on ξ. Sincefor a symplectic form ω on a manifold M the space of tame almost complex structures is inparticular connected, we can define the Chern classes of (M,ω).

Definition 1.10. If J is a tame almost complex structure for the symplectic form ω on M,the Chern classes of (M,ω) are by definition the Chern classes of the complex vector bundle(T M,J).

1.4 Kahler manifolds.

If M is a complex manifold of real dimension 2n, it is possible to define an almost complexstructure on M as follows. Let (Uα,ϕα) be a trivialising cover for the tangent bundleπ : TM→M, that is, ϕα is a fibre-preserving and fibrewise complex linear diffeomorphismπ−1(Uα)→Uα×Cn, and write an element of TM in the form (q,v), with q ∈ M and v ∈ TqM.If q ∈Uα, denote by ϕα(q) the restriction of the diffeomorphism ϕα to the fibre over q: then

ϕα(q) is a complex isomorphism π−1(q)∼=→ Cn and we can define

J(q,v) = (q,ϕα(q)−1J0ϕα(q)(v)),

where J0 is the matrix

(0 −11 0

)and 1 denotes the n× n identity matrix. This definition

does not depend on the choice of trivialising neighbourhood. In fact, if q ∈ Uα ∩Uβ and gαβare the transition matrices of T M, we have

ϕβ(q)−1J0 ϕβ(q)(v) = ϕα(q)−1 ϕα(q)ϕβ(q)−1 J0 ϕβ(q)(v)

= ϕα(q)−1gαβ(q)J0 ϕβ(q)(v)

= ϕα(q)−1J0 gαβ(q)ϕβ(q)(v)

= ϕα(q)−1J0 ϕα(q)(v),

where gαβ(q) commutes with J0 because it is an element of GL(n,C). So J is a well definedalmost complex structure on M.

Definition 1.11. Whenever an almost complex structure J on an arbitrary manifold can berepresented by the matrix J0, with respect to some local coordinates, it is called integrable.

Every complex manifold can also be endowed with a Hermitian metric, denoted by〈 , 〉: this is by definition bilinear, C-linear in the first slot, C-antilinear in the secondone and satisfies the additional two properties

(i) 〈v,w〉 = 〈w,v〉;


(ii) 〈v,v〉 ≥ 0, with equality if and only if v = 0.

The imaginary part of such a metric, which can be written as

Im〈v,w〉 =− i2(〈v,w〉−〈v,w〉),

is skew-symmetric and nondegenerate. Moreover, it is compatible with the standard almostcomplex structure J defined above. To see this, choose local coordinates with respect towhich J is represented by J0 and 〈v,w〉 = vT H w. Then

〈Jv,Jw〉 = (J0v)T H J0w = vT JT0 H J0w

= vT HJT0 J0w = vT H w

= 〈v,w〉,

where we have used the fact that H is hermitian, hence it commutes with J0, and JT0 J0 = 1.

Definition 1.12. If M is a complex manifold, endowed with a Hermitian metric whose imag-inary part is closed, then it is called a Kahler manifold; in particular, it is symplectic with anintegrable almost complex structure.

On the other hand, one can define a Kahler manifold as a symplectic manifold (M,ω) witha compatible integrable almost complex structure J. This definition is equivalent to the oneabove: one can show that M is then complex, using the fact if some local transition functionscommute with the matrix J0, then they must be elements of the complex general linear group,and the inner product

〈v,w〉 = ω(v,Jw)+ iω(v,w)

defines a hermitian metric on M whose imaginary part coincides with ω by definition.

1.5 4-manifolds and the intersection form.

Poincare duality on a closed oriented manifold of dimension n sets up an isomorphism PDbetween the groups Hk(M;Z) and Hn−k(M;Z). If N is a k-dimensional submanifold of M,we can associate to it a k-dimensional homology class [N] and an (n− k)-cohomology classPD[N], that is,

PD : Hk(M;Z)∼=−→ Hn−k(M;Z)

[N] 7−→ PD[N]

In particular, if the dimension of M is 4, Poincare duality yields an isomorphism H2(M;Z) ∼=H2(M;Z). Under this isomorphism, the intersection product of two homology classes [N1]and [N2] corresponds to evaluation of the cup product PD[N1]∪PD[N2] on the fundamentalhomology class of M, that is

[N1] · [N2] = 〈PD[N1]∪PD[N2], [M]〉

1.6. PSEUDO-HOLOMORPHIC CURVES. 5

Definition 1.13. The above product is denoted by QM([N1], [N2]). The form QM is a sym-metric and bilinear form on H2(M;Z) (equivalently, H2(M;Z)), which goes under the nameof intersection form. Since QM vanishes on torsion elements, we can regard it as a form onthe free part of H2. By choosing a basis of H2(M;Z)/Tor we can represent QM by a matrix,called intersection matrix.

In particular, the matrix associated to the intersection form QM always has determinant±1. This can be seen by choosing a basis x1, . . . ,xn of the free part of H2(M,Z). We denoteby x∗i the corresponding dual basis of the free part of H2(M,Z) with respect to the Kroneckerproduct

H2(M)×H2(M) −→ Z

(α,x) 7−→ 〈α,x〉

and set yi := PD(x∗i ). Then QM(xi,y j) is the identity matrix and QM(xi,x j) is equal to thematrix of the coordinate change from the y to the x basis. The latter has determinant ±1,since it is invertible over Z, hence so has QM(xi,x j).

1.6 Pseudo-holomorphic curves.

Let M be a smooth closed manifold, J an almost complex structure on M.

Definition 1.14. A pseudo-holomorphic curve on M is a map from a compact Riemannsurface Σ, with complex structure j, to M, such that

d f · j = J ·d f : T Σ −→ T M.

(i.e., d f is a complex linear bundle map).

Remark. If M is endowed with a symplectic structure ω, and J and ω are compatible, thensmoothly embedded pseudo-holomorphiccurves are also symplectically embedded (cf. Lemma3.3 and 4.29).


Chapter 2

Construction of symplecticmanifolds.

2.1 Introduction.

We have seen that a symplectic manifold with an integrable almost complex structure isa Kahler manifold and that, on the other hand, any complex manifold admits a Hermitianmetric whose imaginary part, if closed, is a symplectic form.

The main classical examples of symplectic manifolds were indeed Kahler manifolds, inparticular, nonsingular complex-projective algebraic varieties: in fact, complex projectivespaces admit a Kahler structure and this induces a Kahler structure on any complex submani-fold.

The most fruitful results in the direction of constructing new symplectic manifolds wereactually achieved in the attempt of finding examples of non-Kahler symplectic manifolds.

2.2 Thurston’s construction.

Definition 2.1. A symplectic fibration is by definition a fibration π : M→B, where the fibreis a compact symplectic manifold (F,σ) and the structure group consists of symplectomor-phisms of the fibre. A symplectic form ω on the total space M is said to be compatible withthe fibration π if each fibre is a symplectic submanifold of (M,ω).

A necessary condition for M to admit a compatible form ω is the existence of a coho-mology class a ∈ H2(M) which restricts to the cohomology class of the symplectic form oneach fibre. Thurston has shown that, if M is compact and the base manifold B is a symplecticmanifold, this condition is also sufficient.

Theorem 2.2 (Thurston). Let π : M→B be a symplectic fibration with compact total spaceM, symplectic fibre (F,σ) and connected symplectic base (B,β). For b ∈ B, denote by ib the

7

8 CHAPTER 2. CONSTRUCTION OF SYMPLECTIC MANIFOLDS.

inclusion of the fibre Fb = π−1(b) in M. Fix identifications ϕb : Fb∼=→ F and denote by σb the

pullback ϕ∗bσ of the symplectic form on F. Suppose that there is a class a ∈ H2(M) such that

i∗ba = [σb] for all b. Then for large enough K ∈ R , the manifold M admits a symplectic formwhich is compatible with the fibration and represents the class K[π∗β]+a.

Proof. The first step of the proof consists in finding a closed symplectic form τ which repre-sents the class a and restricts to the canonical symplectic form in each fibre. This is done bychoosing a closed two-form τ0 representing a. Let (Uα,φα) be a trivialising atlas for thefibration π : M→B. In particular, Uα is an open covering of B and by passing, if necessary,to a refinement, we may assume that each Uα is contractible and that the covering is locallyfinite. For each trivialisation φα : π−1(Uα)→Uα×F, denote by σα the pullback of σ along theprojection Uα ×F→F . By the assumption on a, the form φ∗

α(σα − τ0) is exact, hence thereexist one-forms λα such that φ∗α(σα − τ0) = dλα. Choose a partition of unity ρα subordinateto Uα and define

τ = τ0 +∑d((ραπ)λα).

Then τ is closed, represents the class a and restricts to σb in each fibre. It is nondegenerate onthe subbundle ker(dπ), hence for large enough K the form ωK = Kπ∗β+ τ is nondegenerateon M.

Example 2.3. Let E be a complex rank (n + 1)-bundle over a connected symplectic base B.Let M = P(E) be the projectified bundle, with projection ρ : P(E)→B, and lE ⊂ ρ∗E thecanonical line subbundle. Observe that the induced bundle map lE→B has fibres diffeomor-phic to L, the canonical line bundle over CPn. The first Chern class c1(l∗E) ∈ H2(P(E)) hasthe properties required of the class a in the theorem. In fact,

c1(l∗E)|P(Ep) = c1(l

∗E |P(Ep)) = c1(L

∗)

and the latter coincides with the class of the standard Kahler form on CPn.

Example 2.4. We are going to give a first example of a symplectic, non-Kahler manifold.Consider the T 2-bundle over S1 defined as

[0,1]×T 2/∼, (0,y1,y2) ∼ (1,y1 + y2,y2),

that is, the ends of [0,1] are indentified and the corresponding fibres glued with a Dehn twist.Then cross with S1 to obtain the T 2 bundle over T 2

N = [0,1]×S1×T 2/ ∼, (0,x,y1,y2) ∼ (1,x,y1 + y2,y2).

The projection over T 2 is given by [t,x,y1,y2] 7→ (t,x) ∈ T 2 = S1 ×S1. The manifold N hasby construction odd first Betti number, namely b1(N) = 3, hence it cannot be Kahler: Hodgedecomposition, in fact, implies that all odd Betti numbers of a Kahler manifold must be even.On the other hand, as a bundle over T 2 it admits a section

s(t,x) = [t,x,1,1], where (1,1) = (e2πi0,e2πi0) ∈ S1 ×S1.

2.2. THURSTON’S CONSTRUCTION. 9

The structure group may be reduced from the group of orientation preserving diffeomor-phisms of T 2 to the group of symplectomorphisms with respect to the standard Kahler form,that is, from Diff+(T 2) to Symp(T 2,σ). Let β denote the Poincare dual of s(T 2) in N: then[β] ∈ H2(N) has the properties of the cohomology class a in the statement of Theorem 2.2and this implies that N admits a symplectic structure. The symplectic form is the one inducedby dt ∧dx+dy1∧dy2.

Remark. The manifold M has an almost complex structure JM, with respect to which π isJM-holomorphic. This is constructed in [11] by choosing a metric, denoting by H ⊂ T M thesubbundle of orthogonal complements to the fibres of π with respect to this metric and settingJ|H equal to the pullback of some almost complex structure on B, compatible with β. Sinceeach fibre already has a canonical almost complex structure, JM can be uniquely defined onT M by linearity. By construction, the form ωK tames JM for K large enough. Therefore, giventwo forms ωK = Kπ∗β+τ and ωK′ = K ′π∗β+τ′, with τ and τ′ closed two-forms representingthe class a and restricting to a canonical symplectic form in each fibre, we can interpolatebetween them. Each ωs = sωK +(1− s)ωK′ will be nondegenerate (and obviously closed),hence symplectic. If K = K ′, all these forms will be cohomologous and ωK and ωK′ will beisotopic. Moreover, by Moser stability (Theorem 1.5), they will be strongly isotopic, henceisomorphic.

In particular, Theorem 2.2 implies the following result for surface bundles.

Corollary 2.5. Let F be a compact oriented Riemann surface of genus different from one.Then the total space of any oriented fibration with fibre F and compact symplectic base Badmits a compatible symplectic form.

Remark. This result applies for instance to S2-bundles over compact symplectic manifolds.In that case, if s is a smooth section of the given bundle, we can always choose the symplecticform ω on M so that s is in fact a symplectic section. To see this, choose a closed two-form βon M such that [β] = PD[s(B)] ∈ H2(M,Z). Then for each fibre F of M with inclusion i wehave

〈i∗β, [F ]〉 = 〈[β]∪PD[F], [M]〉 = [s(B)]· [F ] = 1.

On the other hand, if we let ωF correspond in each fibre to the standard symplectic form witharea 1 on S2, then

R

F i∗β = 1 =R

F ωS2 , hence i∗[β] = [ωS2 ] (because H2(M,Z) ∼= Z). Then[β] satisfies the conditions imposed on the class a by the statement of Theorem 2.2 and by themethod of Thurston, it is possible to construct a closed two-form η′ such that i∗(η′) = ωF .The form η := η′−π∗s∗η′ has the same properties and moreover s∗η = 0. Also according toThurston, the form ωK := Kπ∗ωB +η is then, for K large enough, symplectic and compatiblewith the fibration. It is immediate that, with respect to ωK , s(B) is a symplectic submanifoldof M.


2.3 Symplectic blow-up

2.3.1 Blow-up at a point.

The operation of blowing up can be performed in the symplectic category as follows. Webegin by describing the blow up of a symplectic manifold at a point. Let (M2n,ω) be aclosed symplectic manifold, x ∈ M. Then an open neighbourhood U of x can be identified(symplectically) with a neighbourhood V of the origin in Cn, with its standard symplecticstructure ω0. Let p : L→CPn−1 be the tautological line bundle over CPn−1. Then there isa projection Φ : L→Cn, which is a diffeomorphism outside Φ−1(0). The latter is a copy ofCPn−1 and coincides with the zero section of p. Let V := Φ−1(V ).

LΦ−−−−→ Cn

py

CPn−1

Definition 2.6. The blow-up of M at the point x is defined as the sum

M := M−U ∪∂U V

The manifold Φ−1(0) ∼= CPn−1 ⊂ V is called the exceptional divisor of the blow-up. Inthe case of a manifold of dimension 4, the exceptional divisor is a sphere. It is embedded inV ⊂ M as the zero section of the tautological line bundle over CP1, hence its normal bundlecoincides with the vertical bundle p∗(L). From this we see that the square of this sphere isequal to −1. In fact:

〈c1(νMΦ−1(0)), [Φ−1(0)]〉 = 〈c1(νV Φ−1(0)), [Φ−1(0)]〉= 〈p∗c1(L), [Φ−1(0)]〉= 〈c1(L), [CP1]〉= −1.

Sometimes it will be useful to have the following description of blow-up at one point.

Lemma 2.7. If M has dimension 2n, then its blow-up at one point is diffeomorphic to theconnected sum M#CPn, where CPn denotes the n-dimensional complex projective space withopposite orientation.

Proof. By definition, we have

M#CPn = (M −B2n)∪∂B2n (CPn −B2n).

Notice that CPn −B2n ∼= CPn −[1 : · · · : 0] admits a complex line bundle structure overCPn−1, with projection

[x0 : · · · : xn] 7→[

x0

xn: · · · :

xn−1

xn

]

2.3. SYMPLECTIC BLOW-UP 11

in local homogeneous coordinates in the neighbourhoodUn = xn 6= 0. Then the zero section

x0 = 0∼= CPn−1 ⊂CPn has normal bundle isomorphic to the tautological line bundle L overCPn−1 and CPn −B2n may be identified with a tubular neighbourhood of the zero section inL. The claim follows by comparing with the definition of M.

2.3.2 Symplectic form on the blow-up.

Let ω0 and τ0 denote the standard Kahler forms on Cn and CPn−1, respectively. The con-struction of a symplectic form ω on M takes place in three steps.

(i) Prove that the form Φ∗ω0 + εp∗τ0 is nondegenerate on L for all ε > 0 : let JL be thecanonical almost complex structure on L. Then Φ∗ω0(v,JLv) ≥ 0 and p∗τ0(v,JLv) ≥ 0for all vectors v (since p and Φ are JL-holomorphic). Moreover, if v is a nonzero vectorand Φ∗ω0(v,JLv) = 0, then necessarily v ∈ T (Φ−1(0)), since Φ∗ is an isomorphismelsewhere. But then p∗τ0(v,JLv) cannot vanish, because p∗ : T (Φ−1(0))→TCPn−1 isan isomorphism. Hence (Φ∗ω0 + εp∗τ0)(v,JLv) > 0 for all nonzero vectors and for allε > 0.

(ii) Construct a symplectic form ρ on V which equals Φ∗ω0 near ∂V . Notice that the formp∗τ0 is exact outside Φ−1(0), i.e., there exists a one-form β such that p∗τ0 = dβ onL−Φ−1(0). Define a form ρ on V as follows:

ρ =

Φ∗ω0 + εp∗τ0 on Φ−1(0)

Φ∗ω0 + εd(µβ) on V −Φ−1(0),

with µ a smooth function which equals one near Φ−1(0) and zero near ∂V . Since Φ∗ω0

is nondegenerate on V −Φ−1(0), ρ will be nondegenerate if we choose ε sufficientlysmall. Equivalently, since JL tames Φ∗ω0 on V −Φ−1(0) and the taming condition isopen, JL tames ρ.

(iii) Since ρ = Φ∗ω0 outside a neighbourhood of Φ−1(0) in the interior of V , we may defineω on M as follows: Define

ω =

ω on M−Uρ on V .

Remark. With the form we have defined, the exceptional divisor Φ−1(0) is a symplecticsubmanifold of M. In fact, ω|T Φ−1(0) = εp∗τ0 and therefore

〈[ω|T Φ−1(0)]n−1, [Φ−1(0)]〉 = εn−1〈τn−1

0 , [CPn−1]〉 > 0.

2.3.3 Blow-up along a submanifold.

As to blow-up along a submanifold, consider a symplectic embedding i : N−→M. Let Edenote the normal bundle of this inclusion: since M and N carry almost complex structures,


the vector bundle E also admits a complex structure, defined by the short exact sequence0−→TN−→T M|N−→E−→0. In other words, E may be identified with the symplectic or-thogonal bundle of N, which carries a symplectic bundle structure, hence a complex structureas well. With respect to this structure we consider the projectivisation P(E). We choose atubular neighbourhood U of N in M: this can be symplectically identified with a neighbour-hood V of the zero section of E. Let lE be the tautological line bundle over P(E), denoteby p the bundle projecton lE−→P(E) and by Φ the projection lE−→E, so that we have thefollowing diagram

lEΦ−−−−→ E

py

yπ

P(E)ρ−−−−→ N

Definition 2.8. Let V := Φ−1(V ): we define the blow up of M along N to be the manifold

M := M−U ∪∂U V .

Then a symplectic form on M may be defined as in the case of blow-up at a point: onealso has to take care, though, of the normal direction. The precise construction in carried outin [19].

Example 2.9. It is also shown in [19] how blow-up along a symplectic submanifold can beused to generate examples of simply connected, symplectic, non-Kahler manifolds. Oneconsiders for example the T 2-bundle N over T 2 of Example 2.4: this can be symplecticallyembedded in CP5 with the standard symplectic structure. By blowing up CP5 along N weget a symplectic manifold, which is still simply connected because the fundamental group isinvariant under blow-up. This manifold isn’t Kahler: this can be detected, for example, bylooking at the Betti numbers. It turns out, in fact, that b3 = 3 (cf. Example 2.4).

2.4 Symplectic connected sum.

Gompf has shown in [9] how to construct the connected sum of two manifolds along diffeo-morphic submanifolds, under the assumption that an orientation reversing diffeomorphism ofthe normal bundles of the submanifolds is given. Furthermore, he has proved this to be anoperation in the symplectic category, namely: if we assume all manifolds and embeddings tobe symplectic, then the result of the operation will also admit a symplectic structure.

Let ji : N→Mi, i = 1,2 be disjoint codimension two embeddings of closed oriented mani-folds and denote by Ni the images ji(N) and by νi their normal bundles in Mi. Suppose more-over that there exists a fibre-orientation reversing bundle isomorphism ψ : ν1→ν2. This con-dition can be also expressed by saying that the normal bundles have opposite Euler classes.By identifying each νi with a small tubular neighbourhood Vi of Ni and composing ψ withthe diffeomorphism z 7→ z/ ‖ z ‖2 in each fibre, we obtain an orientation preserving diffeo-morphism φ : V1 −N1→V2 −N2.

2.4. SYMPLECTIC CONNECTED SUM. 13

Definition 2.10. The symplectic connected sum of M1 and M2 along N is the manifold

(M1 −N1)∪φ (M2 −N2),

and is denoted by M1#φM2.

Its diffeomorphism type depends on the choice of the embeddings and of the orienta-tion reversing bundle isomorphism ψ. Isotopic embeddings, though, still give rise to diffeo-morphic manifolds, as do bundle isomorphisms which are connected by a fibre-preservingisotopy.

Now suppose that the manifolds Mi and N and the embeddings ji are symplectic: thenM1#φM2 admits a symplectic structure. Assume for simplicity that the given embeddingshave symplectically trivial normal bundles. Choose trivialisations νi

∼= N ×C. By the Sym-plectic Neighbourhood Theorem there exist symplectic embeddings fi : N ×Dε→Mi suchthat fi(N ×0) = Ni. Let Vi = fi(N ×Dε), so that Vi −Ni = fi(N × (Dε −0)). Considerthe following automorphism of the punctured disk:

ρ : Dε −0 → Dε −0(r,θ) 7→ (

√ε2 − r2,−θ).

(2.1)

This is in fact a symplectomorphism with respect to the standard area form ω = r dr dθ,since in polar coordinates on the punctured disc one has:

ω(r0,θ0)(∂r,∂θ) = r0

and

ρ∗ω(r0,θ0)(∂r,∂θ) = ωρ(r0,θ0)(ρ∗(r0,θ0)∂r,ρ∗(r0,θ0)∂θ) =

= ω(√

ε2−r20,−θ0)

− r0√

ε2 − r20

∂r,∂θ

=√

ε2 − r20 ·

r0√ε2 − r2

0

= r0.

Let φ be defined by the commutative diagram

N × (Dε −0) id×ρ−−−−→ N × (Dε −0)

f1

yy f2

V1−N1 −−−−→φ

V2 −N2

Then φ is a symplectomorphism and the manifold

M = (M1 −N1)∪φ (M2 −N2)

admits a symplectic structure.The following is a standard example of the dependence of the diffeomorphism type of a

symplectic sum from the choice of gluing diffeomorphism.


Example 2.11. We consider trivial torus bundles over T 2 and S2 and perform the symplecticconnected sum along two fibres T 2×pt, which are symplectically embedded tori of squarezero. Topologically we obtain the sum

(T 2 ×T 2 −T 2 ×D2)∪T 2×∂D2 T 2 ×D2

that is, we can think of the symplectic sum as obtained from the trivial T 2-bundle over T 2

by cutting out a tubular neighbourhood of one fibre and gluing it back in by an orientationpreserving diffeomorphism of the boundary. Thus the manifold obtained is again a T 2-bundleover T 2, but from the classification of such bundles (compare with [6]), we know that differ-ent gluing diffeomorhisms give rise to different diffeomorphism types of the total space ofthe bundle. So if we choose standard framings for our symplectic sum, the gluing diffeomor-phism will be just the identity map and we will recover the trivial bundle. We could make adifferent choice of framing, though: for instance, with a suitable choice of a twisted framing,it is possible to construct the torus bundle of Example 2.4.

2.5 Branched coverings.

2.5.1 Definitions.

Definition 2.12. Let M and N be smooth n-dimensional manifolds. A smooth map f : M→Nis called a k-fold branched covering if it is a smooth proper map with critical set B ⊂ N suchthat

(i) f |M − f−1(B) : M− f−1(B)→N −B is a covering map of degree k,

(ii) for every point p ∈ f−1(B) there are coordinate neighbourhoods

U,V∼=→ C×Rn−2 around f (p) and p, respectively, on which f is given by

(z,x) 7→ (zm,x)

for some integer m called the branching index of f at p.

The branching index is a local invariant, hence it is constant on each connected componentof f−1(B).

The critical points of f are the points in f −1(B) at which the differential of f fails to besurjective: we will denote the set of these points by C. By definition, f |M − f −1(B) is anordinary covering, hence in particular a local diffeomorphism, thus Tp f is an isomorphismat all points of M − f−1(B). Given p ∈ C and a neighbourhood U of p as in (ii), we seethat C ∩U can be identified locally with 0×Rn−2 ⊂ C×Rn−2. This shows that C is acodimension 2 submanifold of M. Moreover, f |C is an immersion.

For each p ∈ B, the fibre f−1(p) is discrete. Since f is proper, f −1(p) is also compact,hence it must be a finite set, say q1, . . . ,qn. Then for every qi there exist neighbourhoodsUi,Vi such that f |Vi : Vi→Ui is given by (z,x) 7→ (zmi ,x). We may assume that Ui = U for alli and that the Vi’s are all disjoint.

2.5. BRANCHED COVERINGS. 15

In fact, by shrinking U we may assume thatF

Vi = f−1(U). For otherwise, denoting byU j the 1

j -neighbourhood of p, we would find s j ∈ f−1(U j) but not contained in Vi for any iand thus obtain a sequence s j with f (s j) converging to p. By passing to a subsequence wemay assume that s j is convergent, say s j → s with f (s) = p by continuity. But then wewould have s ∈ f−1(p), hence s = qi ∈Vi for some i, which is a contradiction. Therefore wemay assume that the Vi’s are exactly the connected components of f −1(U).

For each p and U as above, the restriction f |Vi − f−1(B) : Vi − f−1(B)→N −B is an mi-fold covering, whereas f |Vi ∩ f−1(B) : Vi ∩ f−1(B)→U ∩B is a diffeomorphism. Thus wehave shown the following result:

Proposition 2.13. The restriction f | f −1(B) : f−1(B)→B of a branched covering to thepreimage of the branching set is an unbranched covering.

We now briefly recall the classification of branched coverings with a given branching setin the differentiable setting.

Lemma 2.14. Let f : M→N be a branched covering and suppose that the branching set B isan embedded submanifold of N: then f is determined, up to diffeomorphism, by the subgroupf∗(π1(M− f−1(B))) < π1(N −B).

Remark. Notice that B = f (C) is in general just an immersed submanifold.

Proof. Let νB be a tubular neighbourhood of B in N. Then νB has a D2-bundle structure overB and its boundary ∂νB a circle bundle structure. The structure group is in both cases O(2).The preimages f−1(νB) and f−1(∂νB) inherit analogous bundle structures over f −1(B).

The bundle projection σ : f −1(∂νB)→ f−1(B), for example, is defined as follows. For q ∈f−1(∂νB) with image p ∈ ∂νB, we choose a neighbourhood U of p as in (ii) of the definitionof branched coverings and assume furthermore that, denoting by π the bundle projection∂νB→B, we have π−1(U ∩ B) ⊂ U . Suppose V1 is the connected component of f −1(U)containing q: then f |V1 ∩ f−1(B) : V1 ∩ f−1(B)→U ∩B is a diffeomorphism, so there existsa unique r ∈V1 ∩ f−1(B) such that f (r) = π(p). Set σ(q) = r.

The S1-bundle structure is uniquely determined by f |M − f −1(B) and this in turn deter-mines the D2-bundle that fills it (the correspondence is 1− 1 because the structure group isO(2)).

2.5.2 Cyclic branched coverings.

Definition 2.15. A branched covering f : M→N with branching set B ⊂ N is called cyclic iff |M− f−1(B) is a cyclic covering in the usual sense, that is, the cyclic group Zk acts properlydiscontinuously on M− f −1(B) and N −B ∼= (M− f−1(B))/Zk.

By the previous Lemma and the classification of cyclic unbranched coverings, f is com-pletely determined by the branching set B and a surjective homomorphism ρ : π1(N−B)→Zk.Notice that since Zk is abelian, the epimorphism ρ factors through the abelianisationH1(N − B) of π1, hence f is also uniquely determined by a surjective homomorphismτ : H1(N −B)→Zk.


Example 2.16. Let the cyclic group Zk act on M by orientation preserving diffeomorphisms.If N denotes the set of fixed points of this action, assume that every component of N hascodimension 2 and that Zk acts freely on M −N. Then M0 = M/Zk is a manifold and thequotient map p : M→M0 is a cyclic branched covering, which maps N diffeomorphicallyonto p(N) ⊂ M0. There are local coordinates (z,ξ) ∈ C×Rn−2 defined in a neighbourhoodof N, with N described locally by z = 0, and (z′,ξ′) ∈ C×Rn−2, defined in a neighbourhoodof p(N), with p(N) described locally by z′ = 0. With respect to these coordinates, the map phas the local description z′ = zn, ξ′ = ξ.

We now turn our attention to the problem of the existence of (cyclic) branched coveringswith a given branching set. Hirzebruch shows in [15] how to construct such coverings underthe condition that the branching index divides the Poincare dual of the branching set.

Proposition 2.17. Suppose we are given a codimension 2 embedding of compact, orientedmanifolds B ⊂ N and an integer k ∈ Z satisfying the condition that k |PD[B]N ∈ H2(N;Z).Then we can construct a k-fold cyclic branched cover of N with branching set B.

Proof. Let E ′→N be the complex line bundle with first Chern class equal to PD[B]N . Sincek |PD[B]N , there exists x ∈ H2(N;Z) such that PD[B]N = kx. Let E be the complex linebundle over N satisying c1(E) = x. Then in particular E⊗k ∼= E ′. Notice that, if H1(N) = 0,then H2(N) has no torsion and the isomorphism class of E is uniquely defined.

There exists a smooth section s : N→E ′ with the following properties:

(i) s vanishes on B;

(ii) s is everywhere nonzero on N −B;

(iii) s is transverse to the zero section of E ′.

It suffices to construct such a section in a tubular neighbourhood U of B. The conditionc1(E ′) = PD[B]N is saying that the normal bundle of B in N is isomorphic to the pull-backE ′|B of E ′ along B. Since E ′ is a complex bundle, we find a cover Vα of B and a trivialisationof E ′|B of the form

E ′|Vα

∼=−→Vα ×C.

By identifying U with a tubular neighbourhood of the zero section of E ′, we get a cover(Uα,Φα) of U with Φα(Uα) ∼= Rn−2 ×C. If we compose Φα with projection onto thecomplex coordinate, we get a map fα : Uα→C such that N ∩Uα = f−1

α (0). Let gαβ := fα f−1β

and denote by E ′ the bundle defined by the transitions functions gαβ with respect to thetrivialising cover Uα. By construction, c1(E ′) = e(E ′) = PD[B] = kx.

Locally, there are sections sα : Uα→Uα ×C, p 7→ (p, fα(p)). By definition of the tran-sition functions, these sections glue together to give a global smooth section s of E ′, whichvanishes on B and is everywhere nonzero on U −B. In fact, s can be extended to a sectionwhich is everywhere nonzero on N −B. Moreover, s(N) intersects the zero section of E ′

transversely: this is also easily checked locally.Define

τ : E −→ E ′ ∼= E⊗k

v 7−→ v⊗·· ·⊗ v,


and consider the diagramE

τ−−−−→ E ′ ∼= E⊗k

πy

yπ′

N≡−−−−→ N

We claim that M := τ−1s(N) ⊂ E is a smooth manifold and that f : M→N, where f is therestriction of the bundle projection π, is the desired cyclic k-fold branched covering of N,branched along B.

To see that the first claim holds, it is enough to observe that τ is transverse to s(N).As to the second one, notice first of all that since π is a bundle map, it is open and closed.

The same holds for f , which moreover has compact fibres and is therefore proper. Of thedefinition of branched coverings, condition (i) is immediately verified: in fact, if p /∈ B, thens(p) 6= 0 and f−1(p) = τ−1(s(p)) consists of precisely k points (all k-fold roots of s(p)).Clearly, f is cyclic. Condition (ii) is easily verified for τ |M : M→s(N); one just needs totake coordinate neighbourhoods around the zero section and let z be the fibre coordinate.This also proves that (ii) holds for f : it suffices to observe that τ and f , when restrictedto M, only differ by the diffeomorphism s. By construction, the upstairs branching set isf−1(B) = τ−1(s(B)) and f | f−1(B) : f−1(B)→B is a diffeomorphism.

Remark. We can thus construct a cyclic branched covering as a submanifold of the totalspace of some complex line bundle over the base manifold, or rather of the associated spherebundle. This admits a symplectic form (by Thurston’s method) which, as we will see in thenext section, restricts to a symplectic form on the branched covering.

2.5.3 Symplectic structures on branched coverings.

Proposition 2.18. Let f : M→N be a k-fold branched covering, branched along a codimen-sion 2 smooth submanifold B ⊂ N. Assume that N admits a symplectic form ω such that B isin fact a symplectic submanifold. Then M admits a symplectic form ω, which coincides withf ∗ω outside an arbitrarily small neighbourhood of the upstairs branching set f −1(B).

Remark. A proof of this result using symplectic cuts is given in [10].

Proof. Let Bi be the connected components of f −1(B). The branching index on eachcomponent is constant, say ki. The situation described in the statement is illustrated by thecommutative diagram

S

i Bi = f−1(B) −−−−→ My

y f

B −−−−→ N

Denote by C the critical set of f , that is C =S

ki≥2 Bi. Since f |M−C : M−C→N −B is alocal diffeomorphism, the form f ∗ω is symplectic along M −C. If ki ≥ 2, then f |Bi : Bi→Bis a connected (unbranched) covering, so the form σi := ( f |Bi)

∗(ω|T B) is symplectic alongBi, i.e., the manifold (Bi,σi) is symplectic.


Let νB be a tubular neighbourhood of B in N with its D2-bundle structure over B (withprojection p). This induces a D2-bundle structure (over f −1(B)) on f−1(νB) such that eachconnected component νBi of f−1(νB) is a tubular neighbourhood of Bi, with projectionpi : νBi→Bi making the following diagram commutative

νBif |νBi−−−−→ νB

pi

yyp

Bif |Bi−−−−→ B

We are going to use the following result, which is contained in [9, Lemma 2.2]. We referto the same paper for details of the proof.

Lemma 2.19. Let (B,ωB) be a closed symplectic manifold, π : E→B an SO(2)-vector bundleover B. Denote by E0 and S = E0 ∪∂E0 E0, respectively, the associated D2- and S2-bundlesover B, with E0 denoting the bundle E0 with opposite orientation. Let i0 : B→S be the zerosection of E0. Then there is a closed 2-form η on S with i∗0η = 0 and η restricting to asymplectic form of area 1 on each fibre. Moreover, η can be chosen so that η|E0 extends to aclosed form on E that is symplectic on each fibre.

Sketch of the proof. Let β be a closed 2-form on S, representing PD[i0(B)] ∈ H2(S,Z). Thenfor each fibre F of S with inclusion iF we have

〈i∗F [β], [F]〉 = [i0(B)] · [F] = 1 = 〈[ωF ], [F ]〉

with ωF denoting the standard symplectic form with area 1 on S2. Hence i∗F [β] = [ωS2 ] ∈H2(F,Z) ∼= Z, the last isomorphism given by evaluation on the class [F].

Since its fibres are compact Riemann surfaces, the bundle map π has in fact a symplecticfibration structure and we may apply the first part of Thurston’s theorem in order to get aclosed 2-form η′ on S which restricts to ωF on each fibre. Let η := η′−π∗i∗0η′. Then η alsorestricts to the standard symplectic form on each fibre and moreover i∗0η = 0.

In order to prove the extension property, one needs to make some appropriate choices inthe proof of the theorem of Thurston.

We return to the proof of Proposition 2.18 and apply the above lemma to the manifold Bi

and the normal bundle pi : Ei→Bi of Bi in M. This yields a closed 2-form ηi on Ei, which issymplectic on each fibre of Ei.

By construction, [η′i] = [β], hence ηi is exact away from Bi and there exists a 1-form γi on

Ei −Bi such that η = dγi on Ei −Bi.At points of Bi, the form f ∗ω is symplectic on T Bi, whereas ηi is nondegenerate in normal

direction. This implies that f ∗ω + tηi is symplectic along Bi for all values of t smaller thanor equal to a constant ti.

Since f ∗ω + tiηi is nondegenerate along Bi, it will be symplectic along a small tubularneighbourhood ν0Bi of Bi. We may assume that ν0Bi is contained in νBi. In fact, f ∗ω+ tηi

will be symplectic along ν0Bi for all t ≤ ti.


Let λi be a smooth bump function in radial direction on νBi, which is identically 1 nearν0Bi and equals 0 near the boundary of νBi.

Define

ω :=

f ∗ω+ tiηi on ν0Bi

f ∗ω+ ∑tid(λiγi) on M−∪ν0Bi.

Then ω is smooth by construction. Moreover, by compactness it is symplectic onM−∪ν0Bi for a choice of ti sufficiently small. Hence by reducing the ti’s, if necessary,ω will be symplectic on M.

Finally, we observe that since ηi|T Bi = 0, the following holds:

ω|T Bi = f ∗ω|T Bi + tiηi|T Bi = f ∗(ω|T B) = σi,

i.e., (Bi,σi) embeds symplectically in (M, ω).

We now focus on cyclic branched coverings, constructed as in the proof of Proposition2.17. In this situation we would like to regard M as a symplectic submanifold of the totalspace of the S2-bundle S associated to the line bundle E.

Assume that N admits a symplectic form ω. Let ρ be the projection S→N ωK = Kρ∗ω+ηthe symplectic form on S, with η a closed 2-form restricting to a form of area 1 in each fibreand vanishing along the zero section of E0. By Lemma 2.19, we may assume that η|E0

coincides with the restriction of a closed 2-form on E which is symplectic in each fibre. Thenwe have the following:

Proposition 2.20. The restriction of the form ωK to M is again a symplectic form, i.e., M isa symplectic submanifold of (S,ωK).

Proof. Let i denote the inclusion of M in S = E0 ∪∂E0 E0. By compactness we may in factassume that i maps M to the interior of E0. Then we can write

ωK |T M = i∗ωK

= i∗(Kρ∗ω+η)

= K f ∗ω+ i∗η.

At points of f−1(B), the tangent space to M splits as

T M| f−1(B)∼= νM f−1(B)⊕T f−1(B).

We take a closer look at νM f−1(B): since by definition f = ρ i, we see that νM f−1(B) ∼=f ∗(E|B) ∼= (i∗ρ∗E)| f−1(B). As in Lemma 2.19, we may assume that η|E0 extends to a closedform on E that is symplectic on each fibre and from this we may conclude that it is non-degenerate on i∗π∗E| f−1(B) (equivalently, symplectic in the fibres of νM f−1(B), which areisomorphic to the fibres of E). On the other hand, f ∗ω is nondegenerate on the symplecticcomplement of the vertical bundle with respect to i∗η, namely T f−1(B), hence K f ∗ω + i∗ηwill be nondegenerate on TM| f−1(B) if K is sufficiently large, say K ≥ K1. Hence it will be

nondegenerate in a tubular neighbourhood ν f −1(B) of f−1(B), again for all K ≥ K1.


On the complement of ν f −1(B), the form f ∗ω is symplectic, therefore, by compactness,ωK |T M is symplectic for K sufficiently large, say K ≥ K2. Hence for K larger than both K1

and K2, we conclude that ωK |TM defines a symplectic form on M.

Chapter 3

Chern classes of symplecticmanifolds.

3.1 Introduction.

We have already seen that the Chern classes of a symplectic manifold (M,ω) are well definedas the Chern classes of a tame almost complex structure. We would now like to answer thequestion: when do two symplectic forms induce the same Chern classes?

Definition 3.1. Two symplectic forms ω0 and ω1 on the same manifold M are called defor-mation equivalent if there exists a smooth family of symplectic forms ωt on M connectingω0 and ω1.

Remark. The symplectic forms ωt need not be cohomologous. If that is the case, we say thatω0 and ω1 are isotopic and that is a much stronger condition: on a closed manifold M, in fact,it implies that (M,ω0) and (M,ω1) are symplectomorphic (compare Definition 1.4).

The answer to our initial question is that the Chern classes of a symplectic manifold(M,ω) only depend on ω up to deformation equivalence.

Lemma 3.2. Let ω0 and ω1 be symplectic forms on the 2n-dimensional manifold M. If theyare deformation equivalent, then they induce the same Chern classes, that is,

ci(M,ω0) = ci(M,ω1) for all i = 1, . . . ,n = dimC M

Proof. Let ωt be a smooth family of symplectic forms connecting ω0 and ω1. Then thereexists a smooth family of bundle isomorphisms ϕt : T M→TM such that ϕ∗

t ωt = ω0 for allt ∈ [0,1] (cf. [20, Ex. 2.4]). Then ϕ1 : (T M,ω0)→(TM,ω1) is an isomorphism of symplecticvector bundles, hence an isomorphism of the underlying complex vector bundles.

In handling the constructions introduced in the previous chapter, one would like to beable to speak of their Chern classes. It is necessary to understand, though, to which extent

21

22 CHAPTER 3. CHERN CLASSES OF SYMPLECTIC MANIFOLDS.

this can be done unambiguously. Here is one result which relies on the contractibility of thespace of almost complex structures on a given symplectic manifold and which will be usefulin handling this problem.

Lemma 3.3. Let (M,ω) be a symplectic manifold, N a symplectic submanifold of M. Givenany tame almost complex structure J0 on the complement of N in M, one can find a tamealmost complex structure JM on M which coincides with J0 outside a tubular neighbourhoodof N and which is adapted to N, in the sense that T N is JM-invariant and JM|T N is again analmost complex structure, tame with respect to ω|T N .

Proof. Let E be the normal bundle of N in M and U a tubular neighbourhood of N, symplec-tomorphic to a tubular neighbourhood of the zero section of E. Then U admits an almostcomplex structure JU which is adapted to N: compare also with the Remark following Exam-ple 2.4.

The space J of almost complex structures on U −N is contractible, i.e., the identity mapis homotopic to the constant map sending any almost complex structure J to J0, or rather itsrestriction to U −N. Let F : J ×I→J be the given homotopy. We may assume that F(J,0) = Jand F(J, t) = J0 for all t ≥ 1− ε for some small positive ε.

If we let 0≤ t ≤ 1 denote the radial coordinate in U , so that p ∈U may be written as (x,v)in some bundle chart, with x ∈ N and ‖ v ‖= t, we can define an almost complex structure onJM on M as follows:

JM(x,v) = F(JU , t)(x,v) for (x,v) ∈UJM ≡ J0 on M−U

Then JM|T N = J0|T N is again a tame almost complex structure and JM = J0 outside U .

3.2 Symplectic sphere bundles.

3.2.1 Chern classes of projective bundles.

Let ρ : S→N be an S2-bundle with compact symplectic base (N,β) and fibre F . Recall thatwe consider on S the symplectic form ωK = Kρ∗β + η, with K ∈ R sufficiently large andη a closed 2-form, restricting to the standard symplectic form of area 1 on each fibre of ρ:compare with the Remark following Theorem 2.2. This definition involves some choices andtwo symplectic forms of this kind will not in general be symplectomorphic. One can show,though, that they induce the same Chern classes.

Given a tame almost complex structure J on N, following [11] we define an almost com-plex structure JS on S. Let H be the subbundle of T S given pointwise as follows: for a pointp in a fibre F of S, Hp is defined as the symplectic orthogonal complement of TpF , that is,

Hp = v ∈ TpS | ωK(v,w) = 0 for all w ∈ TpF.

This definition makes sense because each fibre is a symplectically embedded submanifold ofS. Then ρ∗|H : H→TN is an isomorphism and ωK(v,w) = 0 for all v ∈ kerρ∗ and w ∈ H. Let

3.2. SYMPLECTIC SPHERE BUNDLES. 23

j be the standard complex structure on S2. Then we define JS to be the pullback ρ∗J on Hand j on kerρ∗, the vertical bundle. Finally, we extend by linearity, that is: we write a vectorv ∈ T S as vF + vH and define JSv = jvF +ρ−1

∗ Jρ∗vH .It was already shown that two forms ωK and ωK′ are deformation equivalent, hence by

Lemma 3.2, they induce the same Chern classes. Alternatively, one can show that they tamethe same almost complex structure, namely JS.

Lemma 3.4. Any symplectic form ωK as in Theorem 2.2 tames the almost complex structureJS.

Proof. Let v = vF + vH ∈ TS, v 6= 0.Then

ωK(v,JSv) = ωK(vF + vH , jvF +ρ−1∗ Jρ∗vH)

= ωK(vF , jvF)+ωK(vH ,ρ−1∗ Jρ∗vH)

= η(vF , jvF )+Kρ∗β(vH ,ρ−1∗ Jρ∗vH)+η(vH ,ρ−1

∗ Jρ∗vH)

= η(vF , jvF )+Kβ(ρ∗vH ,Jρ∗vH)+η(vH ,ρ−1∗ Jρ∗vH).

Notice that η restricts in each fibre to the canonical symplectic form on S2: the latter tamesj, so η(vF , jvF ) > 0 unless vF = 0. If vH=0 we have thus ωK(v,JSv) = η(vF , jvF) > 0. IfvH 6= 0, the term Kβ(ρ∗vH ,Jρ∗vH) is positive because ρ∗|H is an isomorphism and β tames J,whereas the term η(vH ,ρ−1

∗ Jρ∗vH) may be negative, but by compactness it is bounded belowby a constant. Therefore Kβ(ρ∗vH ,Jρ∗vH)+ η(vH ,ρ−1

∗ Jρ∗vH) is positive for K sufficientlylarge.

The Chern classes of (S,ωK) are therefore well-defined as Chern classes of (S,JS). Withrespect to this almost complex structure, we have the following complex vector bundle iso-morphism:

T S ∼= ρ∗T N ⊕kerρ∗.

In particular, we then have the Whitney formula

c(T S) = ρ∗c(T N)∪ c(kerρ∗).

Assume from now on that S = P(E) is obtained by projectifying a complex rank 2 bundleE over N and let lE ⊂ ρ∗E be the tautological line bundle over S.

lE ⊂ ρ∗E −−−−→ Ey

yπ

S = P(E)ρ−−−−→ N

If we set c1(l∗E) =: ξ, there exists a ring isomorphism

H∗(S;Z)∼=→ H∗(B;Z)[ξ]/〈ξ2 +π∗c1(E)ξ+π∗c2(E)〉, (3.1)

see [3, p. 270]. Let P(Ep) ∼= CP1 be the fibre of ρ over a point p ∈ N and let ip denote theinclusion P(Ep) → S. The bundle lE restricts over one such fibre to the canonical line bundle


L over CP1. Since c1(L∗) generates H2(CP1;Z) ∼= Z and is compatible with the orientation(i.e., 〈c1(L∗), [CP1]〉= 1), we conclude that i∗pξ = i∗pc1(l∗E) = c1(L∗) must be equal to the classof the standard symplectic form of area 1 on CP1. Hence, according to Thurston’s theorem,in the definition of ωK we may choose η so that it represents the class ξ.

Consider the quotient bundle E (1) = ρ∗E/lE .

Lemma 3.5. We have the following isomorphism of complex line bundles:

kerρ∗ ∼= E(1)⊗ l∗E .

Proof. One needs to show that E (1)⊗ l∗E fits into the short exact sequence

0−→E(1)⊗ l∗Eα−→ TP(E)

β−→ ρ∗TN−→0 (3.2)

which defines kerρ∗, with the morphism β induced by ρ∗.In order to check exactness, we need to define α. Restrict first to a fibre of S over a point

p ∈ N, say P(Ep) ∼= CP1: then E(1) and l∗E restrict to the canonical quotient bundle Q and thedual L∗ of the tautological line bundle over CP1. We have there already an isomorphism (see,for example, [3, p. 281])

Q⊗L∗ ∼= TCP1.

According to [23], the above isomorphism, followed by the inclusion TP(Ep)⊂ TP(E)|P(Ep),extends to a well-defined bundle map

α : E(1)⊗ l∗E→TP(E).

Exactness of the sequence may now be checked fibrewise. After choosing a Riemannianmetric, the canonical quotient bundle Q may be identified with the orthogonal complementof L. Thus Q⊗L∗ ∼= Hom(L,Q) ∼= Hom(L,L⊥). Given a point l ∈ P(Ep), the fibre of L atthis point may be identified with l itself and the fibre of Q with l⊥. At this level, then, αgives an isomorphism Hom(l, l⊥)∼= TlP(Ep). On the other hand, given (p, l) ∈ P(E), we alsohave identifications lE (p,l) ' l and Q(p,l) ' l⊥. Since T(p,l)P(E) splits as TlP(Ep)⊕TpN, thesequence at (p, l) has the form

0−→Hom(l, l⊥)α−→ TlP(Ep)⊕TpN

β−→ ρ∗T N−→0 (3.3)

and by definition of the morphisms α and β it is obviously exact.

Hence the Chern classes of S may be expressed as follows:

c(T S) = ρ∗c(T N)∪ c(E(1)⊗ l∗E) (3.4)

= ρ∗c(T N)∪ c(ρ∗E ⊗ l∗E) (3.5)

= ρ∗c(T N)∪2

∑i=0

ci(ρ∗E)(1+ξ)2−i, (3.6)


where the second equivalence follows from the exact sequence

0−→lE−→ρ∗E−→E(1)−→0 (3.7)

by tensoring with l∗E , which gives

0−→C−→ρ∗E ⊗ l∗E−→E(1)⊗ l∗E−→0, (3.8)

thus showing that c(E(1)⊗ l∗E) = c(ρ∗E ⊗ l∗E).

Example 3.6. If the dimension of N equals 4, for example, and E is a complex line bundleover N, the Chern classes of the projectified bundle S = P(E ⊕C) are given by:

c1(S) = ρ∗(c1(N)+ c1(E))+2ξ,c2(S) = ρ∗(c1(N)∪ c1(E)+ c2(N))+2ρ∗c1(T N)ξ,c3(S) = 2ρ∗c2(N)ξ.

(3.9)

3.2.2 Cyclic branched coverings.

Let f : M→N be a k-fold cyclic branched cover, constructed as in the previous chapter, withbranching set B ⊂ N. Recall that M = τ−1s(N) ⊂ E and f = π|M, with π : E→N a complexline bundle over N satisfying PD[B] = k c1(E), s a section of E⊗k

Mi−−−−→ E

f

yyπ

N≡−−−−→ N

and τ : E→E⊗k =: E ′ the k-fold tensor product map. The projection E ′→N is denoted by π′.We have seen in Proposition 2.20 that if the manifold N admits a symplectic form β, then

M may be regarded as a symplectic submanifold of the S2-bundle associated with the linebundle E, that is S = P(E ⊕C), with respect to the symplectic form ωK = Kρ∗β+η.

We would like to compute the Chern classes of M with respect to the restriction of thissymplectic form, i.e., with respect to the restriction of a tame almost complex structure on S,say JS, which is adapted to M: the existence of such an almost complex structure is guaran-teed by Lemma 3.3, applied to the symplectic embedding of M in S. Then by the symplectictubular neighbourhood theorem, this defines an almost complex structure in a tubular neigh-bourhood of the zero section of E. This almost complex structure restricts to an almostcomplex structure on M. To summarise, we may obtain the Chern classes of M, regarded asa symplectic submanifold of S, from the relation

π∗(c(E)∪ c(T N))|M = c(T E)|M = c(T M)∪ c(νE M),

where νEM denotes the normal bundle of M in E and is described up to isomorphism by thefollowing lemma.

Lemma 3.7. The normal bundle of s(N) in E ′ is isomorphic to the vertical bundle (π′)∗E ′

and the normal bundle of M in E to its pullback τ∗(π′)∗E ′|M = f ∗E ′.


Proof. The first claim is immediate because s is a section.To prove the second one, we need to use the transversality of τ and s(N). Namely, there

is a map φ, induced by τ∗, fitting into the commutative diagram

νE M ∼= TE|M/TMφ−−−−→ TE ′|s(N)/T s(N) ∼= νE′s(N)

yy

Mτ|M−−−−→ s(N)

It is defined in the following way:

φ[(x,v)] = [(τ(x),τ∗(v))]

for all [(x,v)] ∈ T E|M/T M, which means x ∈ M, v ∈ TxE and [(x,v)] = [(x,v′)] if and only ifv− v′ ∈ TxM. Then φ covers τ by definition.

If [w] ∈ TyE ′/Tys(N), with y = τ(x) ∈ s(N), by transversality of τ and s(N) we may writew as u+v, with u ∈ Tys(N) and v ∈ τ∗TxM. Hence [w] = [v] = [τ∗s] = φx[s] for some s ∈ TxMsuch that τ∗s = v. This shows that φ is fibrewise surjective. Then, by dimension arguments,it is fibrewise an isomorphism.

Thus φ is a bundle map and the universal property of the pullback implies the claim.

We have the following sequence of isomorphisms:

νEM ∼= τ|∗MνE′s(N) ∼= τ|∗M(π′|s(N))∗E ′ ∼= π|∗ME ′ ∼= ( f ∗E)⊗k.

Hence the total Chern class of M can be computed using the relation

c(T E)|M = c(T M)∪ (1+ k f ∗c1(E)). (3.10)

By induction one can now prove the following result.

Proposition 3.8. Let f : M→N be a symplectic k-fold cyclic branched covering, obtained asabove from the complex line bundle π : E→N. Then the Chern classes of M are given by

cr(M) = f ∗cr(N)+(1− k)r

∑i=1

(−1)i−1ki−1 f ∗cr−i(N)∪ f ∗ci1(E).

Proof. First of all, from the relation c(T E) = π∗c(N)∪π∗c(E) we get for the single Chernclasses the expression

ci(TE) = π∗ci(N)+π∗(ci−1(N)∪ c1(E)).

If we denote by i the inclusion of M in E, equation (3.10) implies that

i∗c1(TE) = ci(M)+ k f ∗c1(E)∪ ci−1(M).


To finish the proof we compute

cr+1(M) = i∗cr+1(T E)− k f ∗c1(E)∪ cr(M)

= f ∗cr+1(N)+ f ∗cr(N)∪ c1(E)− k f ∗c1(E)∪ f ∗cr(N)+ (3.11)

−k(1− k) f ∗c1(E)r

∑i=1

(−1)i−1ki−1 f ∗cr−i(N)∪ f ∗c1(E)i

= f ∗cr+1(N)+(1− k) f ∗cr(N)∪ c1(E)+

+(1− k)r

∑i=1

(−1)iki f ∗cr−i(N)∪ f ∗c1(E)i+1

= f ∗cr+1(N)+(1− k)r

∑i=0

(−1)iki f ∗cr−i(N)∪ f ∗c1(E)i+1

= f ∗cr+1(N)+(1− k)r+1

∑j=1

(−1) j−1k j−1 f ∗c(r+1)− j(N)∪ f ∗c1(E) j,

where (3.11) follows from inductive hypothesis. So the Proposition is proved.

Example 3.9. (Compact Riemann surfaces as branched coverings of S2.) Let a be the gene-rator of H2(S2) compatible with the orientation. Our aim is to construct a 2-fold branchedcovering of S2, branched at 2g + 2 points. Consider the complex line bundle π : L→S2, de-fined by c1(L) = (g + 1)a. Then c1(L⊗2) = 2(g + 1)a = PD[(2g + 2)pt]. By applyingthe cyclic branched covering construction, we get a compact surface Σg ⊂ L and a 2-foldbranched covering f : Σg→S2. Then one can compute

c1(Σg) = f ∗c1(S2)− f ∗c1(L) = (1−g) f ∗a

and〈c1(Σg), [Σg]〉 = (1−g)〈 f ∗a, [Σg]〉 = 2(1−g)〈a, [S2]〉 = 2(1−g)

that is, Σg is exactly the compact Riemann surface of genus g and the covering we haverecovered is the well-known quotient map induced by rotation of Σg by π around a verticalaxis with 2g+2 fixed points.

3.2.3 Other submanifolds: sections.

We are considering the S2-bundle ρ : S→N, where S = P(E ⊕C) for a complex line bundle Eover the symplectic manifold (N,β). Again we may also regard S as obtained by gluing alongtheir boundary two copies of a closed disc subbundle of E, namely S = E0∪∂E0 E0, where E0

has opposite orientation to E0.There are embeddings i+ : N→S and i− : N→S, corresponding to the zero sections of E

and E, respectively. We denote their images in S by N+ and N−. Then the normal bundleof each of these sections coincides with the vertical bundle in E or E and we have νSN+ =ρ∗E|N+ and νSN− = ρ∗E|N− .


Notice that N+ and N− are symplectic submanifolds of (S,ωK = Kρ∗β + η). In fact,ωK |T N+ = Kρ∗β|T N+ +η|T N+ and since ρ∗β is nondegenerate on T N+, also ωK will be non-degenerate for K sufficiently large. The same argument applies to the restriction of ωK toT N−.

Remark. The restriction of the bundle map ρ to either section is not a symplectomorphism,but it is compatible with the almost complex structure, i.e., the restriction of its differentialρ∗ to T N+ or T N− is a complex isomorphism. Hence c(N+) = ρ∗|N+c(N) and similarly forN−.

3.2.4 Donaldson’s theorem.

The following theorem is taken from [5].

Theorem 3.10. Let (M,ω) be a compact symplectic manifold with integral cohomology class[ω] ∈ H2(M,Z). Then for every sufficiently large integer λ there exists a connected codimen-sion 2 submanifold Nλ ⊂ M which represents the Poincare dual of λ[ω].

Remark. Writing [ω] ∈ H2(M;Z) already denotes the choice of an integral lift of [ω] ∈H2(M;Z).

The integrality condition on [ω] is not a restrictive one for our purposes. In fact, we mayalways assume that it is satisfied.

Lemma 3.11. Given a symplectic manifold (N,β), there exists an integral symplectic form βon N, inducing the same Chern classes as β.

Proof. First we approximate β by a closed rational form β′. In order to do this, choose abasis u1, . . . ,um for H2(N;Z) and 2-forms α j ∈ Ω2(N) representing the element of the basis,that is, [α j ] = u j. Then there exist coefficients λ j ∈ R such that [β] = ∑m

j=1 λ jα j ∈ H2(N;R).Now consider the form

β′ = β+m

∑j=1

(r j −λ j)α j, r j ∈ Q.

By choosing the r j’s to be rational we obtain a rational form. In fact,

[β′] = [m

∑j=1

r jα j] =m

∑j=1

r ju j ∈ H2(N;Q).

The differences (r j − λ j) can be made arbitrarily small and for a sufficiently small pertur-bation the form β′ is still symplectic. Moreover, since we can obviously linearly interpolatebetween β and β′, the two forms induce the same Chern classes and hence the same Chernnumbers. Now choose a positive integer x ∈ Z>0 such that x[β′] ∈ H2(N;Z) and set β := xβ′.By construction, the form β is symplectic and represents an integral cohomology class. It ishomotopic to β′, hence also to the original form β, so it induces the same Chern classes.

An immediate consequence of the above lemma is that given a symplectic sphere bundle(M,ωK) with symplectic base (N,β), we may replace ωK by ωK := K ′ρ∗β+ η, where K ′ is

3.3. CHERN CLASSES OF BLOW-UP. 29

an integer larger than K, β is an integral symplectic form on N, satisfying the condition in thelemma, and η has been chosen among the representatives of c1(l∗E). Replacing ωK with thenew symplectic form does not affect the Chern classes, and [ωK ] is integral by construction.

Corollary 3.12. We can always assume the form ωK on M to be integral. In this situationDonaldson’s theorem implies that for sufficiently large λ ∈Z, there exist symplectic submani-folds Xλ ⊂ M satisfying the relation PDM[Xλ] = λ [ωK ].

The Chern classes of such submanifolds Xλ can be computed from the relation

c(M)|Xλ= c(Xλ)∪ c(νMXλ)

= c(Xλ)∪ (1+PDM[Xλ]|Xλ)

= c(Xλ)∪ (1+λ[ωK]|Xλ).

See Section A.2 in the Appendix for computations in one particular instance.

3.2.5 Branched coverings as submanifolds.

Proposition 2.20 shows the following: suppose S is an S2-bundle over a symplectic base(N,β) with projection ρ : S→N, obtained by compactifying the complex line bundle E→N,that is, S = P(E ⊕C). If there exists a symplectic submanifold B of N such that PDN [B] =k c1(E) for some k∈N, then there also exists a symplectic submanifold M of S with PDS[M] =ρ∗c1(E⊗k): it is the k-fold branched covering of N along B. This is a much more elemen-tary and explicit way of finding submanifolds of S2-bundles over a symplectic base than theapplication of Donaldson’s theorem. By numerical reasons, though, in order to include suchsubmanifolds in the blow-up systems of Chapter 4, we need to find symplectically embeddedcurves in some 4-dimensional symplectic manifold, realising a multiple of a 2-dimensionalhomology class with negative square. In this way we come across the problem of solvingsymplectic singularities: we do not know how this can be achieved within a fixed homologyclass.

3.3 Chern classes of blow-up.

This section is essentially an adaptation of the results contained in [18] in a smooth topolog-ical setting.

We consider the symplectic embedding i : N→M, where N and M are closed symplecticmanifolds, and denote the corresponding normal bundle by E. Let M be the blow up of Malong N. Then we have the commutative diagram

P(E)j−−−−→ M

ρE

yy f

Ni−−−−→ M

which is referred to as the blow-up diagram.


One fundamental remark is the following: recall that M was defined as M−U∪∂U V , withV a neighbourhood of the zero section of the canonical bundle lE , and consider the inclusionP(E) ⊂ M. It is in fact an inclusion P(E) ⊂ V ⊂ lE , which takes P(E) to the zero sectionof lE , where lE is regarded as a complex line bundle over P(E) itself. This implies that thenormal bundle of P(E) in M is isomorphic to lE . In other words, we have the following shortexact sequence:

0−→TP(E)−→T M|P(E)−→lE−→0 (3.12)

(Notice that TP(E) is a complex vector bundle).The map f in the blow-up diagram also deserves a few words. It is defined as follows:

f =

id on M−UπΦ on V

where Φ denotes the projection lE→E. With this definition, f is a diffeomorphism outsideP(E) and, indeed, the blow-up diagram is commutative.

3.3.1 Some cohomological lemmas.

We start by proving some general results, which apply in particular to the blow-up situation.The general reference for the following section is [4].

Definition 3.13. For any map f : N(n)→M(m) of smooth, compact, oriented manifolds, onecan define “shriek” homomorphisms

f ! : Hn−p(N,∂N) → Hm−p(M,∂M)

andf! : Hm−p(M,∂M) → Hn−p(N,∂N)

by f ! = PDM f∗(PD−1N ) and f! = (PDN)−1 f ∗PDM, respectively (see [4, p. 368] for more

details).

Definition 3.14. If W is a k-disk bundle over a manifold N of dimension n, with projectionπ : W→N, and i : N→W denotes the inclusion of N in W as zero section, the Thom class ofW is τ = PDW i∗[N]. The Thom isomorphism

H i(N)π∗→ H i(W )

∪τ→ H i+k(W,∂W )

coincides with i!. If i : N→M is a smooth codimension k embedding of manifolds, possiblywith boundaries that intersect transversely, the Thom class of the inclusion is

τMN = PDM i∗[N] ∈ Hk(M).

If we denote by W a tubular neighbourhood of N in M and identify it with a k-disk subbundleof the normal bundle of N in M, the Thom class τM

N is the image of the Thom class of W under

Hk(W,∂W )exc∼= Hk(M,M −W)→Hk(M).


The Euler class of the normal bundle is

χMN = i∗τM

N ∈ Hk(N).

Proposition 3.15 (Excision Lemma). Let i : N→M be an inclusion of smooth compact mani-folds, U := M−N the complement of N in M, with inclusion u : U →M. Suppose λ ∈ H∗(M)satisfies u∗λ = 0. Then there exists β ∈ H∗(N) such that i!β = λ.

Proof. We need to show exactness of the sequence

H∗(N)i!→ H∗(M)

u∗→ H∗(U).

On the other hand, we know that the sequence

H∗(M,U)k∗→ H∗(M)

u∗→ H∗(U)

is exact. Let W be a tubular neighborhood of N and consider inclusions i0 : N→W andi1 : W→M such that i1i0 = i. Then i!0 is the Thom isomorphism. Denote by exc the excisionisomorphism and by k the inclusion (M, /0)→(M,U). Then there is a sequence of isomor-phisms

H∗(N)i!0−→ H∗(W,∂W )

≡−→ H∗(W,W −N)(exc)−1

−→ H∗(M,U).

To prove our claim it will be enough to show that k∗(exc)−1i!0 = i!. Notice that since i! = i!1i!0and i!0 is an isomorphism, we have further reduced our claim to the following

k∗(exc)−1 = i!1

for all elements of H∗(W,∂W ) (and in fact, in this form we will apply the lemma later).We therefore set out to prove that the following diagram is commutative:

H∗(W,∂W )(exc)−1

−−−−→ H∗(M,U)k∗−−−−→ H∗(M)

D−1W

yy≡

H∗(W )i1∗−−−−→ H∗(M)

DM−−−−→ H∗(M)

Let α = [ f ] ∈ H∗(W,∂W ): then f is a cochain of W and it is zero on chains of ∂W . Since(exc)−1 : H∗(W,∂W ) ∼= H∗(M,M −N) is an isomorphism, we may extend f to a cochain fof M which is zero on chains of M −N. Then by construction f = k∗(exc)−1 f , and i∗1 f = fas chains on W .

We can choose chains cW and cM , representing [W ] and [M], respectively, in such a waythat i1∗cW = cM. Then on the (co)chains level we have

DMi1∗( f ∩ c) = DM i1∗(i∗1 f ∩ c) = DM( f ∩ i1∗cW ) = DM( f ∩ cM) = f .

Therefore PDM i1∗PD−1W (α) = [ f ] = k∗(exc)−1(α): this proves our claim and the lemma.


The next is a corollary of the excision lemma which applies to blow-up diagrams.

Corollary 3.16. Consider the blow-up diagram relative to the embedding i : N→M. Denoteby u the inclusion of M −N in M. Assume λ ∈ H∗(M) is such that u∗(λ) = 0. Then thereexists µ ∈ H∗(P(E)) such that ( j)!(µ) = f ∗(λ).

Proof. Let v be the inclusion of M−P(E) in M. Then the following diagram commutes:

M−P(E)v−−−−→ M

f

yy f

M−Nu−−−−→ M

Hence v∗ f ∗(λ) = f ∗u∗(λ) = 0. Applying the Excision Lemma to f ∗(λ) yields the claim.

We would now like to show that for any y ∈ H∗(N), with inclusion i : N→M of codimen-sion 2r and normal bundle E, the following holds:

i∗i!(y) = y∪ cr(E),

where cr(E) denotes the top Chern class (i.e., the Euler class) of the bundle E. This equiva-lence goes under the name of Self Intersection Formula. Notice that if y = 1 ∈ H0(N), thenthe formula is easily verified:

i∗i!(1) = i∗PDMi∗[N] = i∗τMN = e(E) = 1∪ cr(E)

with τMN and e(E) denoting the Thom class and the Euler class, respectively.

Lemma 3.17 (Self-intersection formula). For any y ∈ H∗(N),

y∪ cr(E) = i∗i!(y).

Proof. Let W be a tubular neighbourhood of N in M, with inclusions i0 : N→W and i1 :W→M, so that i1i0 = i. Let α be the first morphism in the short exact sequence

H∗(W,∂W )α→ H∗(W )→H∗(∂W ).

Suppose N has codimension 2r in M and denote by τ∈H2r(W,∂W ) the Thom class PDW i0∗[N].Notice that i∗0ατ = cr(E) ∈ H2r(N).

We claim that α = i∗1i!1. In fact, if we choose [ f ] ∈ H∗(W,∂W ), then f extends to acochain f on M which is zero on chains of M −N (cf. Proposition 3.15). By definition,i∗1[ f ] = α[ f ] ∈ H∗(W ), hence it is enough to show that i!

1[ f ] = [ f ]. But

i!1[ f ] = PDMi1∗PD−1W [ f ] =

PDM i1∗(i∗1[ f ]∩ [W ]) = PDM([ f ]∩ [M]) = [ f ].

To finish the proof we just need to carry out one last computation:

i∗i!y = i∗0i∗1i!1i!0y = i∗0α(π∗y∪ τ) =

i∗0(π∗y∪ατ) = y∪ cr(E).


The Self Intersection Formula can be applied to the map j in the blow-up diagram to ob-tain the following corollary. Let ξE denote the first Chern class of the dual of the tautologicalline bundle lE .

Corollary 3.18. Suppose y ∈ H∗(M) is such that ( j)∗(y) = −ξE y for some y ∈ H∗(P(E)).Then y = ( j)! y+λ with ( j)∗λ = 0.

Proof. We have remarked in the previous section that the normal bundle of the inclusionj : P(E)→M is the tautological line bundle, whose first (top) Chern class is −ξE , hence bythe previous lemma we have that ( j)∗( j)! y = −ξE y for all y ∈ H∗(P(E)).

Rewrite y as ( j)! y+ y− ( j)! y and take λ = y− ( j)! y. Then

( j)∗λ = ( j)∗(y− ( j)∗( j)! y) = ( j)∗y− ( j)∗( j)! y = −ξE y+ξE y = 0.

3.3.2 Remark on the definition of Chern classes of blow-up.

The construction of a symplectic form on the blow-up of a symplectic manifold (M,ω) in-volves several choices and yields forms which are not necessarily isomorphic. Still, we wouldlike to show that the Chern classes of such blown up manifolds are well defined. For this weneed Lemma 3.3: it applies in particular to the blow-up situation and allows us to speak aboutthe Chern classes of the blow-up without ambiguity.

Proposition 3.19. Let M denote the blow-up of a symplectic manifold M along a symplecticsubmanifold N with normal bundle E. Let P(E) be the exceptional divisor of the blow-up. IfJM is a tame almost complex structure on M, there exists a tame almost complex structure onM, adapted to P(E) and coinciding with f ∗JM outside a neighbourhood of P(E).

Proof. By looking at the construction of a symplectic form on the blow-up of a symplecticmanifold in [19], we notice that the inclusion P(E)→M is always symplectic. In fact, theform ω is defined on V as p∗ωK + εΦ∗α, with ω a form on P(E) as given by the Examplefollowing Theorem 2.2 and α a closed 2-form on E. Since p∗ω is nondegenerate on TP(E),for ε sufficiently small ω will be nondegenerate as well. Moreover, since f : M→M is adiffeomorphism away from the exceptional divisor P(E), it follows that f ∗JM is an almostcomplex structure on the complement of P(E) in M. If we apply Lemma 3.3 to the symplecticmanifold M and the symplectic submanifold P(E), we get an almost complex structure as inthe statement.

Corollary 3.20. With respect to an almost complex structure as in Proposition 3.19, thesequence (3.12) gives in fact a complex splitting of vector bundles

j∗TM ∼= TP(E)⊕ lE . (3.13)


3.3.3 The blow-up formula.

From the isomorphism (3.13) of vector bundles over P(E) we get

j∗c(T M) = c(TP(E))∪ c(lE)

hence with 2r denoting the codimension of N in M, we can compute

( j)∗(c(T M)− f ∗c(T M)) = c(TP(E))(1−ξE)−ρ∗E i∗c(T M)

= c(kerρ∗)ρ∗Ec(T N)(1−ξE)−ρ∗

Ec(T N)ρ∗E c(E)

= ρ∗Ec(T N)(1−ξE)c(E ′)−ρ∗

Ec(E)

= ρ∗Ec(T N)

r−1

∑i=0

ci(E(1))(1+ξE)r−i−1(1−ξE)−ρ∗

Ec(E).

We claim that the expression in brackets is of strictly positive degree in ξE . In fact, we mightat most get constant terms (i.e., of degree zero in ξE) out of the terms

r−1

∑i=0

ci(E(1))−ρ∗

Ec(E) =r−1

∑i=0

i

∑k=0

ρ∗Eck(E)ξi−k

E −ρ∗Ec(E)

=r−1

∑i=0

i−1

∑k=0

ρ∗Eck(E)ξi−k

E +r−1

∑k=0

ρ∗Eck(E)−

r

∑k=0

ρ∗Eck(E)

=r−1

∑i=0

i−1

∑k=0

ρ∗Eck(E)ξi−k

E −ρ∗Ecr(E)

=r−1

∑i=0

i−1

∑k=0

ρ∗Eck(E)ξi−k

E +r

∑k=1

ρ∗Ecr−k(E)ξk

E

but this expression is of strictly positive degree in ξE (unless it vanishes).Hence j∗(c(T M)− f ∗c(T M)) = −ξEγ, where

γ = − 1ξE

ρ∗Ec(T N)

r−1

∑i=0

ci(E(1))(1+ξE)r−i−1(1−ξE)−ρ∗

Ec(E)

.

That j∗(c(T M)− f ∗c(T M)) must be of positive degree in ξE can also be seen as follows. LetJM be a compatible almost complex structure in M as in the statement of Proposition 3.19.Outside a tubular neighbourhood U of P(E), the difference c(M)− f ∗c(M) is zero, becausethe almost complex structure there coincides with the pullback of a tame almost complexstructure on M, that is, JM = f ∗JM. In fact, c(M)− f ∗c(M) vanishes on the complement ofP(E) in M, since H∗(U) ∼= H∗(M). Then by Corollary 3.16, one finds γ ∈ H∗(P(E)) suchthat c(M)− f ∗c(M) = j!γ. By application of the Self-Intersection formula we obtain

j∗(c(M)− f ∗c(M)) = j∗ j!γ = −γξE .


We still need to show that the equation j∗(c(M)− f ∗c(M)) = j∗ j!γ = −γξE impliesc(M)− f ∗c(M) = j!γ. By Corollary 3.18 following the proof of the Self-Intersection For-mula, there exists λ such that ( j)∗λ = 0 and c(T M)− f ∗c(T M) = ( j)!γ+λ.

Consider once more the inclusions u : M−N→M and v : M−P(E)→M, so that we havethe commutative diagram

M−P(E)v−−−−→ M

f |M−P(E)

yy f

M−Nu−−−−→ M

Since the sequence H∗(P(E))( j)!

→ H∗(M)v∗→ H∗(M−P(E)) is exact, we have

v∗(c(T M)− f ∗c(T M)) = v∗( j)!γ+ v∗λ = v∗λ.

On the other hand,

v∗(c(T M)− f ∗c(T M)) = c(T (M −P(E))− v∗ f ∗c(T M) = 0

because v∗ f ∗c(T M) = f ∗|M−P(E)u∗c(T M) = c(T (M −P(E))). Hence v∗λ = 0. We claim

that this implies ultimately that λ = 0. By exactness of the sequence

H∗(M,M−P(E))k′∗→ H∗(M)

v∗→ H∗(M−P(E)),

the vanishing of v∗λ implies λ = k′∗µ for some µ∈ H∗(M,M−P(E)). Let W ′ be the preimagein M of a tubular neighbourhood W of N in M. Then W ′ is a tubular neighbourhood ofP(E) and there are inclusions j0 : P(E)→W ′ and j1 : W ′→M, whose composition is equal toj : P(E)→M. We denote by α′ be the first morphism in the short exact sequence

H∗(W ′,∂W ′)α′→ H∗(W ′)→H∗(∂W ′)

and consider the diagram

H∗(M,M−P(E)))(k′)∗−−−−→ H∗(M)

exc

yy j∗1

H∗(W ′,∂W ′)α′

−−−−→ H∗(W ′) ∼= H∗(P(E)).

We have

• j∗1k′∗µ = j∗1λ = 0 ∈ H∗(W ′) since j∗λ = 0 and j∗0 is an isomorphism;

• by commutativity of the diagram, α′ exc(µ) = j∗1k′∗µ = 0;

• injectivity of α′ implies that exc(µ) = 0;

• finally, since the excision map is an isomorphism, we get µ = 0 and therefore λ = 0.


Summarising, we are now able to write down the formula according to which the Chernclasses behave under blow-up.

Theorem 3.21. Let M be a symplectic manifold and let M denote its blow-up along a sym-plectic submanifold N of real codimension 2r. If E denotes the normal bundle of N in M, sothat we have the commutative blow-up diagram

P(E)j−−−−→ M

ρE

yy f

Ni−−−−→ M

the Chern classes of M are well defined and can be expressed as follows:

c(T M)− f ∗c(T M) = (3.14)

j!

[− 1

ξEρ∗

Ec(T N)

r−1

∑i=0

ci(E(1))(1+ξE)r−i−1(1−ξE)−ρ∗

Ec(E)

]

where ξE denotes the first Chern class of the dual of the tautological line bundle over P(E).

In order to gain some confidence about the correctness of this formula, we derive from itthe expression for the first Chern class of the blow-up and compare it with the existing formulafor algebraic manifolds (see, for example, [12]). In the following we drop the subscript Efrom the notation.

Corollary 3.22. The first Chern class of the blow-up of a symplectic manifold M along asubmanifold of codimension 2r has the expression

c1(T M) = f ∗c1(T M)− (r−1) j!1.

Proof. It is enough to observe that j! maps H0(P(E)) to H2(M), so one needs to look forterms of degree one in ξ inside the braces. Recall that ci(E(1)) = ∑i

k=0 ρ∗ck(E)ξi−k. Then itis enough to find in the following expression the terms which are linear in ξ.

r−1

∑i=0

ci(E(1))(1+ξ)r−i−1(1−ξ) =

(1+ξ)r−1(1−ξ)+ c1(E(1))(1+ξ)r−2(1−ξ)+ . . . =

(1+ξ)r−1(1−ξ)+(ξ+ρ∗c1(E))(1+ξ)r−2(1−ξ)+ . . . =

ξ(−1+ r−1+1)+ . . .= (r−1)ξ+ . . .

Taking into account the − 1ξ factor, we obtain

c1(M)− f ∗c1(M) = j![−1

ξ(r−1)ξ

]=−(r−1) j!1.

Remark. Observe that j!1 is exactly the Poincare dual of the exceptional divisor in M: in fact,by definition, j!1 = PD j∗[P(E)].

3.4. SYMPLECTIC SUMS. 37

3.4 Symplectic sums.

3.4.1 Symplectic sum along surfaces with trivial normal bundle.

If two symplectic manifolds are symplectically glued along a codimension 2 submanifoldwith trivial normal bundle, the Chern numbers of the symplectic sum can be expressed interms of the Chern numbers of the original manifolds and those of a trivial projective bundleover the given submanifold. More precisely we have the following statement (cf. [13]):

Lemma 3.23. Let X and Y be symplectic 2n-dimensional manifolds. Suppose N is anothersymplectic submanifold, of dimension 2n−2, and that there exists symplectic embeddings ofN in X and Y with trivial normal bundle. Then we may consider the symplectic connectedsum of X and Y along N, defined as in Section 2.4 and denoted by W := X#NY: this hasChern numbers given by

cI [W ] = cI [X ]+ cI[Y ]− cI[N ×S2]

where I stands for any arbitrary partition of n.

Proof. Let Dε denote the 2-dimensional disk with radius ε. By the Symplectic Tubular Neigh-bourhood Theorem, there exist symplectically embedded neighbourhoods of N of the formN ×Dε both in X and Y , such that N is identified with N ×0 ⊂ N ×Dε. Recall that thesymplectic connected sum of X and Y is obtained by cutting out N from both manifolds andidentifying the tubular shell neighbourhoods N ×D∗

ε = N × (Dε \0) via the map

id×ρ : (p,(r,θ)) 7−→ (p,(√

ε2 − r2,−θ)).

Notice that the circle S0 with radius r0 = ε/√

2 is fixed by ρ, so N×S0 is identified with itself.Denoting by D0 the disc with radius r0, so that ∂D0 = S0, we can describe W (topologically)as

X −N ×D0 ∪N×S0 Y −N ×D0.

Let θ be a top-dimensional form on W . The restriction of θ to X −N ×D0 =: X0 andto Y −N ×D0 =: Y 0 extend to forms θ1 and θ2 on X and Y . In turn, the restrictions of θ1

and θ2 to the tubular neighbourhoods N ×D0 may be glued together to obtain a form θ onN ×D0 ∪N×S0 N ×D0 = N ×S2. Then if we integrate θ on W we get

Z

Wθ =

Z

X0θ|X0 +

Z

Y 0θ|Y 0 (3.15)

=Z

Xθ1 −

Z

N×D0

θ1|N×D0 +Z

Yθ2 −

Z

N×D0

θ2|N×D0

=

Z

Xθ1 +

Z

Yθ2 −

Z

N×S2θ.

Let σi denote the invariant polynomial defined by

det(I + tA) = 1+ tσ1(A)+ . . .+ tnσn(A)


for every square matrix A, and for a partition I = (i1 . . . ir) of n let σI denote the productσi1 . . .σir . Then given connections ∇X and ∇Y on X and Y with curvature tensors K∇X andK∇Y , the polynomials σI(K∇X ) =: θ1 and σI(K∇Y ) =: θ2 represent some product of Chernclasses of X and Y , namely cI(X) and cI(Y ), respectively. By using a partition of unity, wemay assume that ∇X and ∇Y coincide with a product connection over N ×D0. Then theycan be pasted toghether to yield a connection ∇ on W with σI(K∇) =: θ representing cI(W ).Moreover, the product connections over the tubular neighbourhoods N ×D0 give a productconnection ∇ on N × S2 such that σI(K∇) coincides with the form θ obtained by gluing therestrictions of θ1 and θ2 along N ×S0. By applying (3.15) to the situation just described weget

cI [W ] =Z

Wθ =

Z

Xθ1 +

Z

Yθ2 −

Z

N×S2θ

= cI [X ]+ cI[Y ]− cI[N ×S2].

3.4.2 Symplectic sums along tori in dimension 4.

Now let W = X#T 2Y denote the symplectic sum of two 4-dimensional manifolds X and Yalong symplectically embedded tori with square zero. The latter condition means that thefirst Chern class of the normal bundle of each torus, evaluated on its fundamental homologyclass, is zero: that is, 〈c1(ν), [T 2]〉 = 0. Since Kronecker product with the fundamental classdefines an isomorphism between H2(T 2;Z) and Z, we see that the square of T 2 being zerois equivalent to its normal bundle having vanishing first Chern class or, in other words, to itsbeing trivial. Notice that this by no means implies that the torus is homologically trivial.

We may assume that PD−1c1(X) and PD−1c1(Y ) admit representatives which are disjointfrom the tori along which the sum is performed. In fact, let KX = Λ2T ∗X be the canonicalbundle over X . Then c1(K∗

X ) = c1(X) and in particular c1(K∗X |T 2) = c1(T 2)+ c1(νX T 2) = 0

(the first summand is zero because the tangent bundle to any oriented torus is trivial, thesecond because T 2 is embedded with square zero). Hence there exists a section s of K∗

X ,which is nonvanishing over T 2. Let Zs denote the zero locus of s and identify X with the zerosection of K∗

X : then c1(X) = PD[Zs] and Zs ∩T 2 = /0.Hence PD−1c1(X) and PD−1c1(Y ) represent homology classes in W and we may consider

the Poincare duals of these, which are again denoted by c1(X) and c1(Y ). Then we can statethe following result, which is taken from [25].

Lemma 3.24. Let X and Y be symplectic manifolds of dimension 4, containing symplecticallyembedded tori with square zero. If we denote by W their sympletic connected sum along thesetori, we have for the first Chern class of W the expression:

c1(W ) = c1(X)+ c1(Y )−2PD[T 2]. (3.16)

Proof. Consider K∗X , the anticanonical bundle of X . There is a symplectically embedded

neighbourhood N ∼= T 2 ×Dε of T 2 in X . The tangent bundle of X splits over such a neigh-bourhood as L1 ⊕ L2, where L1 and L2 are line bundles corresponding to the tangent and

3.4. SYMPLECTIC SUMS. 39

normal direction to the torus. More precisely, the bundles L1 and L2 represent the pull-backof the bundles parallel and normal to T 2 along the symplectomorphism N ∼= T 2 ×Dε. Thenthe restriction of the anticanonical bundle of X to N has the form

K∗X |N ∼= (Λ2T ∗X)∗ ∼= (Λ2(L∗

1 ⊕L∗2))

∗ ∼= (L∗1 ⊗L∗

2)∗ ∼= L1 ⊗L2.

In particular, a sectionσX : N −→ K∗

X |N ∼= N ×C

can be written as a product σ1X ·σ2

X of sections of L1 and L2, respectively. This implies for thezero sets the relation ZσX = Zσ1

X∪ Zσ2

X. Analogous considerations hold for T 2 ⊂ Y .

So over T 2 × (Dε −0) we also have trivial line bundles L1 and L2. Recall that thesymplectic sum was obtained by performing the identification of (x,z) with (x,ρ(z)), with ρthe symplectic automorphism of the punctured disk defined in (2.1). After identification, then,in the direction parallel to the torus we still have the trivial bundle L1. In normal direction,though, the effect of the identification is that of a connected sum in each fibre. If we denoteby F the fibre of the normal bundle of T 2, we see that L2 may be regarded as the pullback ofthe tangent bundle of this fibre along the projection T 2 ×F→F. After identification, that is,we have the following picture

L2 −−−−→ T (F#F)y

y

T 2 × (F#F) −−−−→ F#F

and there is a contribution to the first Chern class of TW coming from the zero set of asection of the bundle L2. Consider a section σ : F#F→T (F#F): this has zero set Zσ ⊂ Fwhich represents the Poincare dual of the element −2 ∈ H2(F#F) ∼= Z. It pulls back toa section σ of L2, defined by σ(x,y) = (x,y,σ(y)). Then the zero section Zσ is equal toT 2×Zσ ⊂ T 2× (F#F) and represents the class [Zσ] =−2[T 2] ∈ H2(W ). This implies for thefirst Chern class of the symplectic connected sum the equivalence

c1(W ) = c1(X)+ c1(Y )+PDW [Zσ]

= c1(X)+ c1(Y )−2PDW [T 2].


Chapter 4

Symplectic geography.

4.1 Cobordism ring and Chern numbers.

4.1.1 Stable equivalence.

Definition 4.1. A stable complex structure on a real vector bundle ξ is a complex structureon ξ⊕ εk, where ε denotes the trivial line bundle. Two stable complex structures are iso-morphic if there is a stable isomorphism of the defining complex vector bundles. A stablyalmost complex manifold is a smooth manifold M with a stable complex structure on thetangent bundle.

Remark. Recall that a bundle ξ over a manifold is orientable if and only if w1(ξ) = 0, thatis, its first Stiefel-Whitney class vanishes [3, Ex. 12-A]. This, together with the equivalencew1(ξ⊕ ε) = w1(ξ), implies that any stably almost complex manifold is orientable.

Equivalently, a stably almost complex manifold can be defined as a smooth manifoldtogether with an embedding into some large euclidean space whose normal bundle admits acomplex structure. We prove that the two definitions are equivalent in the next lemma.

Lemma 4.2. Let M be a smooth manifold of dimension m. Then the following two conditionsare equivalent:

(i) the tangent bundle T M admits a stable complex structure;

(ii) there exists an embedding M→RN , with m + N ≡ 0(mod2), whose normal bundleν(M,RN) is a complex bundle of rank N−m

2 .

Proof. First of all recall that if ξ is a real vector bundle, the Whitney sum ξ⊕ ξ is isomor-phic to the complexification ξ⊗C of ξ and admits a complex structure defined by J(v,w) =(w,−v). Suppose (i) holds, i.e., that for some integer k the bundle TM⊕εk admits a complexstructure. Consider an embedding of M in Rn for some large n (such an embedding alwaysexists for n > 2m+1) and compose it with the inclusion

Rn→Rn+k = Rn ×Rk.

41

42 CHAPTER 4. SYMPLECTIC GEOGRAPHY.

The resulting embedding M→Rn+k has normal bundle

ν(M,Rn+k) ∼= ν(M,Rn)⊕ εk

and we can write

εn+k ∼= TRn+k|M ∼= TM⊕ν(M,Rn+k) ∼= T M⊕ν(M,Rn)⊕ εk.

By taking the Whitney sum with ν(M,Rn) on both the left and the right hand side we get:

ν(M,Rn)⊕ εn+k ∼= ν(M,Rn)⊕ν(M,Rn)⊕TM⊕ εk.

By the remark at the beginning of the proof, the sum ν(M,Rn)⊕ν(M,Rn) admits a complexstructure and so does T M ⊕ εk by assumption. Hence ν(M,Rn)⊕ εn+k admits a complexstructure (which implies, in particular, that ν(M,Rn) has a stable complex structure). Nowthe required embedding is

M→R2n+k = Rn ×Rk ×Rn,

whose normal bundle ν(M,R2n+k) ∼= ν(M,Rn)⊕ εn+k is indeed a complex bundle.If on the other hand (ii) holds and we are given an embedding M→RN with complex

normal bundle ν(M,RN), then we have

εN ∼= TRN |M ∼= TM⊕ν(M,RN).

By summing with T M we obtain

T M⊕ εN ∼= TM⊕TM⊕ν(M,RN).

On the right hand side, TM ⊕TM again admits a complex structure as the complexificationof T M, whereas ν(M,RN) is a complex bundle by assumption. This implies that TM ⊕ εN

admits a complex structure as well, i.e., (i) holds.

The Chern classes of the bundle ξ may be defined as the Chern classes of the complexbundle ξ⊕ εk and the Chern classes of a stably almost complex manifold M as the Chernclasses of its tangent bundle. This definition only depends on the stable complex structure upto stable isomorphism. In fact, if ξ ⊕ εk and ξ⊕ εh are complex vector bundles definingisomorphic stable complex structures on ξ, that is, such that there is an isomorphism ofcomplex vector bundles

ξ⊕ εk ⊕ εN−k ∼= ξ⊕ εh⊕ εN−h,

then we have that c(ξ⊕ εk) = c(ξ⊕ εh). If M is compact of dimension 2m and stably almostcomplex we may also speak of the Chern numbers of its tangent bundle. In this case, cm[M]does not necessarily coincide with the Euler number.

Example 4.3. Let M = S2n ⊂ R2n+1: the Whitney sum of the normal and tangent bundle ofS2n is trivial, hence S2n admits a stably almost complex structure with cn[S2n] = 0.

Lemma 4.4. Let M and W be stably almost complex manifolds and suppose that there isa smooth embedding of M in W. Let ν denote the normal bundle of M in W. Then ν isorientable.

4.1. COBORDISM RING AND CHERN NUMBERS. 43

Proof. From the decomposition T M⊕ν = TW |M we get w1(T M)+w1(ν) = w1(TW )|M . Themanifolds M and W are stably almost complex, hence orientable, so w1(T M) = w1(TW ) = 0.This implies that w1(ν) must be zero, that is, ν is orientable.

Let M and W be as in Lemma 4.4 and assume furthermore that the embedding of M inW has codimension 1. Let W→RN and M→Rk be the embeddings defining stable almostcomplex structures on W and M. The inclusion of M in RN+1 resulting from the composition

M→W→RN→RN ×R = RN+1

has normal bundleν(M,RN+1) ∼= ν(M,W )⊕ν(W,RN)⊕ ε

which is also a complex bundle: in fact, ν(W,RN) is complex by definition and ν(M,W )⊕ εis a real orientable rank 2 bundle, hence it admits a unique complex structure. In this wayν(M,RN+1) defines a second stably almost complex structure on M, which we call the stablyalmost complex structure induced by W . We say that it is equivalent to the given stable almostcomplex structure if there is a stable isomorphism between ν(M,RN+1) and ν(M,Rk).

On the set of stably almost complex manifolds we can introduce an equivalence relationas follows.

Definition 4.5. Two stably almost complex manifolds M1 and M2 of equal dimension arecalled stably equivalent if they bound a stably almost complex manifold W in such a waythat the stably almost complex structures induced by W on its boundary components, namelyM1 and M2, are equivalent to the given ones. The equivalence classes of n-dimensional stablyalmost complex manifolds with respect to stable equivalence form an abelian group, denotedby ΩU

n . The topological product of manifolds turns the direct sum ΩU = ⊕ΩUn into a graded

ring, called the complex cobordism ring.

The notion of stable equivalence will be from now on referred to as (complex) cobordism.

Definition 4.6. Let E be a U(n)-bundle over a manifold M, with projection π and let E0 bethe associated bundle with fibre D2n ⊂ Cn: its boundary ∂E0 is the bundle associated to Ewith fibre S2n−1 ⊂ Cn. The Thom space of E, denoted by M(E), is the quotient E0/∂E0: itis homeomorphic to the one-point compactification of E. Let BU(n) denote the classifyingspace for the unitary group U(n). If ζn is the universal U(n)-bundle over BU(n), we writeMUn for the Thom space M(ζn). This gives rise to the so called Thom complex MU.

Then by general theory of cobordism (generalized Pontrjagin-Thom theorem), the groupsΩU

n may be identified with the stable homotopy groups πn(MU). The latter may be calculatedby applying homotopy theoretical methods such as Adams’ spectral sequence. See [21] formore details.

Lemma 4.7. The complex cobordism groups ΩUn are zero for n odd and for n = 2m they are

free abelian of rank equal to the number of partitions of m.

Milnor also showed that ΩU is a polynomial ring on even dimensional generators and wasable to produce explicit representatives for these generators, which in particular turn out tobe, together with their inverses, complex projective algebraic varieties. In particular, Milnor’sresult involves so called hypersurfaces of bidegree (1,1).


4.1.2 Hypersurfaces of bidegree (1,1).

Definition 4.8. We call hypersurface of degree (1,1) a subset Hi, j of the product CPi×CP j,defined by

Hi, j = ([w0 : · · · : wi], [z0 : · · · : z j]) ∈ CPi ×CP j |w0z0 + · · ·+wkzk = 0,k = min(i, j).

Lemma 4.9. Hypersurfaces of degree (1,1) are simply connected algebraic varieties of di-mension i+ j−1.

Proof. Suppose i ≤ j. There is an obvious projection

Hi, jπ−→ CPi

([w0 : · · · : wi], [z0 : · · · : z j]) 7−→ [w0 : · · · : wi].

The fibre over a point w = [w0 : · · · : wi] ∈ CPi is given by

π−1(w) = z = [z0 : · · · : z j ] ∈ CP j |w0z0 + · · ·+wizi = 0 ⊂ CP j.

Since not all coefficients in the defining equation can be zero, we have π−1(w) ∼= CP j−1. Itis not difficult to see that the projection π is locally trivial, so we conclude that Hi, j has aCP j−1-bundle structure over CPi and for this reason it is an algebraic variety of the requireddimension.

If we denote by f the inclusion Hi, j→CPi ×CP j and by a and b the 2-dimensional ge-nerators of the cohomology rings of CPi and CP j, respectively, the Poincare dual of Hi, j inCPi ×CP j is a+b, hence in H∗(Hi, j) the following relation holds:

c(Hi, j)∪ f ∗(1+a+b) = f ∗c(CPi ×CP j)

= f ∗(1+a)i+1∪ (1+b) j+1.

Example 4.10. We compute the Chern numbers of H2,2 ⊂ CP2 ×CP2. We start with

c(H2,2)∪ f ∗(1+a+b) = f ∗(1+a)3∪ f ∗(1+b)3

= f ∗(1+3(a+b)+3(a2+b2 +3ab)+9(a2b+ab2)),

where we have disregarded terms which exceed the dimension of H2,2, namely 3. By com-paring the left- and right-hand side of the above equivalence, we get that the Chern classes ofH2,2 are

c1 = 2 f ∗(a+b),

c2 = f ∗(a2 +b2 +5ab),

c3 = 3 f ∗(a2b+ab2).


Recalling that the Poincare dual of H2,2 in CP2 ×CP2 is a + b, we can compute the Chernnumbers. For example,

c31[H2,2] = 〈8 f ∗(3a2b+3ab2), [H2,2]〉

= 〈8(3a2b+3ab2)∪ (a+b), [CP2×CP2]〉= 〈48a2b2, [CP2 ×CP2]〉 = 48.

Similarly,

c1c2[H2,2] = 〈12 f ∗(a2b+ab2), [H2,2]〉= 〈12(a2b+ab2)∪ (a+b), [CP2×CP2]〉= 〈24a2b2, [CP2 ×CP2]〉 = 24

and

c3[H2,2] = 〈3 f ∗(a2b+ab2), [H2,2]〉= 〈3(a2b+ab2)∪ (a+b), [CP2×CP2]〉= 〈6a2b2, [CP2 ×CP2]〉 = 6.

4.1.3 The complex cobordism ring.

Now we can completely describe the structure of the complex cobordism ring.

Proposition 4.11. There exists an isomorphism of graded rings

φ : (Z[x1,x2, . . .], · ,+) −→ (ΩU ,× ,t)xn 7→ [Kn].

Moreover, the manifold Kn in the definition of φ may be chosen in a class of manifolds gene-rated, under Cartesian product and disjoint sum, by complex projective spaces, hypersurfacesof double degree Hi, j ⊂ CPi ×CP j and their inverses.

Let sn denote the unique polynomial in the elementary symmetric functions σ j of vari-ables ti satisfying

sn(σ1, . . . ,σn) =n

∑i=1

tni .

What Milnor shows, then, is that Kn may be taken as 2n-dimensional generator for the ringΩU provided it satisfies the condition

sn[Kn] = 〈sn(c1(Kn), . . . ,cn(Kn)), [Kn]〉

=

±1 if n+1 6= qr for any prime q±q if n+1 = qr for some prime q.

If we assume that a bundle E splits as a Whitney sum of complex line bundles L1 ⊕ . . .⊕Ln, we see that each Chern class ck(E) coincides with the k-th elementary symmetric function


in the variables c1(L1), . . . ,c1(Ln), so that sn(E) = sn(c1(E), . . . ,cn(E)) = ∑i=1 c1(Li)n. In

particular, for a Whitney sum E ⊕F, the characteristic polynomial satisfies then the relationsn(E⊕F) = sn(E)+sn(F). By the splitting principle, this relation holds for arbitrary bundles.If we consider the manifold M = K ×L, we also have that sn(M) = sn(K)+ sn(L) and if Kand L are both of dimension strictly smaller than M, this implies that sn[K × L] = 0. Thecharacteristic number sn[M] does not vanish for all manifolds, though. If M = CPn, forexample, we know that there is an isomorphism TCPn ⊕ ε ∼= (L∗)⊕n+1, with ε denoting thetrivial complex line bundle and L the tautological line bundle over CPn. Then

c(TCPn) = c(TCPn ⊕ ε) = c(L∗)⊕n+1

and

sn[CPn] = 〈sn(c1(L∗), . . . ,c1(L

∗), [CPn]〉

= 〈n+1

∑i=1

c1(L∗)n, [CPn]〉 = n+1.

For n = 1, . . . ,4 the characteristic polynomial sn is given by:

s1 = σ1

s2 = σ21 −2σ2

s3 = σ31 −3σ1σ2 +3σ3

s4 = σ41 −4σ2

1σ2 +2σ22 +4σ1σ3 −4σ4.

Since sn(CPn) = n+1 and s3(H2,2) = −6, we see that it is possible to choose

K1 = CP1

K2 = CP2 (4.1)

K3 = CP3 tH2,2

K4 = CP4.

4.1.4 The notion of C-equivalence.

We would like to describe now the relation between complex cobordism and Chern numbersof (stably) almost complex manifolds.

Definition 4.12. We introduce the quotient Γn, which is obtained from the set of 2n-dimen-sional almost complex manifolds by identifying two manifolds M1 and M2 if and only if theyhave the same Chern numbers. In the literature, this equivalence relation also goes under thename of C-equivalence.

It is shown in [29] by induction that each Γn is a group (in fact, an abelian group), sinceeach element admits an inverse.


Lemma 4.13. For each almost complex manifold M of dimension 2n, the class [M] ∈ Γn

admits an additive inverse.

Proof. As announced, the proof is by induction. In dimension 2 (n = 1) we have −[CP1] =[F2], with F2 a Riemann surface of genus 2.

Now consider the products

CPI = CPi1 ×·· ·×CPik ,

with I = (i1, . . . , ik) a partition of n. By a theorem of Thom (cf. [22, Thm. 16.7]) the matrix(cJ [CPI ]), where I and J run over all possible partitions of n, is non-singular. If M is analmost complex manifold of dimension 2n, we can then find rational coefficients aI such that

cJ [M] = ∑I

aI cJ [CPI ].

Let b be an integer number such that baI ∈ Z for all I. Then

cJ [bM] = ∑I

baI cJ [CPI ]

and the linear combination on the right hand side has integer coefficients (although not nec-essarily positive).

We need to distinguish two cases.Suppose sn[M] = 0. We claim that the coefficient an must be zero. In fact, sn[M] =

∑aIsn[CPI ] and sn[CPI ] = 0 unless I = n, in which case sn[CPn] = n+1. This means thatbM is C-equivalent to a disjoint union of products of complex projective spaces of dimensionstrictly less than n. By inductive hypothesis these products admit an inverse, hence we havean equivalence in Γn, namely

[bM] = ∑I

baI [CPI ].

This shows that [bM] admits an inverse, i.e., there exists N such that −[bM] = [N]. Since wecan write bM as (b−1)MtM, we see that M also admits an inverse, namely

−[M] = [(b−1)M]+ [N].

If sn[M] 6= 0, on the other hand, we may find M′ and integer numbers c and d such thatsn[cM + d M′] = 0, then apply the argument of the previous case. In particular, we maychoose M′ to be a complex projective space or a regular hypersurface, depending on the signof sn[M].

We would like to show that the groups Γn may be identified with the cobordism groupsΩU

n . Then an identity such as −[CP1] = [F2] in the proof of Lemma 4.13 can be geometricallyvisualised as a stably almost complex 3-dimensional manifold N, inducing the (stabilisationof the) standard almost complex structure on its boundary components M1 = S2 t F2 andM2 = T 2, a 2-dimensional torus.


Since each complex cobordism class admits a complex representative (generators do), wecan define maps

φn : ΩUn −→ Γn (4.2)

by sending an element [M] ∈ ΩUn to the class in Γn determined by the Chern numbers of such

a representative of [M]. Since Chern numbers behave additively with respect to disjoint sum,we immediately see that (4.2) is a homomorphism for all n. To show that it is surjective, onejust needs to observe that almost complex manifolds are in particular stably almost complex.The map is well defined and injective because of the following result (see [28]).

Proposition 4.14. Two stably almost complex manifolds are cobordant if and only if theyhave the same Chern numbers.

Proof. We start by showing that cobordant manifolds have the same Chern numbers. Byadditivity, it is enough to show that the Chern numbers of a boundary vanish. So supposewe are considering stably almost complex manifolds W and its boundary ∂W . Let i denotethe inclusion of ∂W in W . If x ∈ Hn(BU ;Z) is a top dimensional product of universal Chernclasses, compatibility of the stable almost complex structures implies that x(∂W ) = i∗x(W ).Let ∂∗ denote the connecting homomorphism of the exact homology sequence of the pair(W,∂W ):

. . .→Hn+1(W,∂W ;Z)∂∗→ Hn(∂W ;Z)

i∗→ Hn(W ;Z)→ . . .

Since [∂W ] = ∂∗[W ], we have

〈x(∂W ), [∂W ]〉 = 〈i∗x(W ),∂∗[W ]〉 = 〈x(W ), i∗∂∗[W ]〉 = 0 (4.3)

by exactness of the sequence above. Hence all Chern numbers of ∂W are zero.To show the converse, consider the composition of the Hurewicz homomorphism

πn(MU) −→ Hn(MU ;Z)

with the Thom isomorphism

Hn(MU ;Z)∼=−→ Hn(BU ;Z).

One gets a homomorphism

τ : ΩUn∼= πn(MU) −→ Hn(BU ;Z).

Recall that Hn(BU ;Z) is generated as a Z-module by the n-dimensional universal Chernclasses cI , with I a partition of n. The kernel of the Hurewicz homomorphism is a finitegroup by a theorem of Serre (see [24]), hence it is trivial becase ΩU

n is torsion-free. Thus wemay regard ΩU

n as a subgroup of Hn(BU ;Z) (in fact, for reasons of rank, it is a maximal freesubgroup).

We have then the homology-cohomology pairing

Hn(BU ;Z)⊗Hn(BU ;Z)→H0(pt;Z) ∼= Z.

4.2. THE GEOGRAPHY PROBLEM. 49

If [M] ∈ ΩUn , the above pairing coincides with the evaluation of the top dimensional charac-

teristic classes on the fundamental class of M. In terms of the generators cI , the pairing sends([M],cI) to cI [M], the I-th Chern number of M. Since the pairing is nondegenerate, eachelement in ΩU

n is determined by its pairing with the generators of Hn(BU ;Z) and this showsthat complex cobordism is completely determined by the Chern numbers.

The topological product of manifolds gives a well-defined operation on Γ =⊕nΓn, whichturns it into a ring. This multiplicative structure coincides under the homomorphisms φn

with the multiplicative structure on the complex cobordism ring ΩU . In view of the iden-tification of the groups Γn with the complex cobordism groups and the identification of thecorresponding ring structures, Proposition 4.11 gives a complete description of the ring Γ,including representatives for the generating equivalence classes.

Proposition 4.15. The C-equivalence ring Γ is a polynomial ring with generators Kn indimension 2n which may be chosen in a class of manifolds consisting of disjoint sums andproducts of complex projective spaces and hypersurfaces of bidegree (1,1).

4.2 The geography problem.

The geography problem for manifolds endowed with an additional structure (which allowsthe definition of their Chern classes) aims at determining which systems of integer numberscan be realised as the system of Chern numbers of such a manifold of suitable dimension.

Let π(n) denote the cardinality of the set of partitions of n. Proposition 4.15 implies forthe Chern numbers of almost complex manifolds the following result, see [14].

Theorem 4.16 (Milnor). A system of π(n) integer numbers occurs as the system of Chernnumbers of a 2n-dimensional almost complex manifold M if and only if it occurs as the systemof Chern numbers of a 2n-dimensional algebraic manifold X, which belongs to the class M ofalgebraic manifolds generated under cartesian product and disjoint sum by projective spaces,hypersurfaces of degree (1,1) and their negatives.

Knowing the Chern numbers of the manifolds generating M and their behaviour underthe operations of cartesian product and disjoint sum, it is in principle possible to write downa set of necessary and sufficient congruence relations for a given system of integer numbersto occur as the system of Chern numbers of an almost complex manifold.

Example 4.17. For K1 and K2 we may choose CP1 and CP2, respectively. Hence we see thatΩ2 is generated by CP1×CP1 and CP2 and their negatives −(CP1×CP1) = CP1×F2, withF2 a compact Riemann surface of genus two, and −CP2 = X(3)#12CP2, where X(3) denotesa hypersurface of degree 3 in CP3. Then a pair of integer numbers (p,q) is realised as thesystem of Chern numbers of an almost complex manifold M of the form a(CP1 ×CP1)tbCP2 if and only if the system

8a+9b = p4a+3b = q

admits a solution in Z2 and this happens if and only if the given numbers satisfy the congru-ence relation p+q ≡ 0 (mod 12).


4.2.1 The theorem of Riemann-Roch.

Some of the congruence relations among Chern numbers of almost complex manifolds arisefrom algebraic geometry, namely the theorem of Riemann-Roch. Let M be an almost complexmanifold of dimension 2n and denote by ci ∈ H2i(M;Z) the Chern classes of its tangentbundle. In the direct product H∗∗(M;Z) of the cohomology groups of M, one writes theseclasses as elementary symmetric functions in variables x j ∈ H2(M;Z) and defines ei to be thei-th elementary symmetric function of the variables ex j −1 [27].

Definition 4.18. Let Tn be the multiplicative sequence of polynomials belonging to the powerseries f (t) = t/(1− e−t) (cf. [22, p. 221]). Then the Todd class of M is defined by

T (M) =∞

∑j=0

Tj(c1, . . . ,c j) ∈ H∗∗(M;Z),

where the first T -polynomials are

T1 =12

c1

T2 =112

(c21 + c2)

T3 =124

c1c2

T4 =1

720(−c4 + c1c3 +3c2

2 +4c21c2 − c4

1)

...

(compare [27] and [16, p. 14]).

In H∗∗(M;Z) we now consider the subring SU , consisting of formal power series over Z

in the elements ei. If α is an element of H∗∗(M;Z)⊗Q, denote by αn the 2n-dimensionalcomponent of α. The Riemann-Roch theorem implies that

〈(z ·T )n, [M]〉 ∈ Z for all z ∈ SU . (4.4)

Since the number 〈(z ·T )n, [M]〉 can be expressed as a linear combination of Chern numbersof M with linear coefficients, this indeed implies some congruence relations among the Chernnumbers of M.

It was conjectured by Atiyah and Hirzebruch [1] and proved by Stong [27] that (4.4)actually implies all relations.

Theorem 4.19 (Stong). The Riemann-Roch theorem

〈(z ·T )n, [M]〉 ∈ Z for all z ∈ SU

gives all relations among the Chern numbers of 2n-dimensional almost complex manifolds.

4.2. THE GEOGRAPHY PROBLEM. 51

As long as we do not impose any connectedness condition, then, we know necessary andsufficient conditions for a given sequence of integer numbers to appear as the system of Chernnumbers of an almost complex manifold (in fact, even Kahler), and we can conclude that theproblem of the geography of almost complex or Kahler manifolds has a well known solution.If we want to consider connected manifolds, though, the problem immediately becomes moredifficult. In fact, one would be tempted to consider connected rather than disjoint sums,but connected sums of almost complex manifolds do not always admit an almost complexstructure. The classical example is given by the connected sum of two copies of CP2 (cf. [2],[17]).

Suppose, for example, that we were interested in determining the geography of connectedalmost complex manifolds. The Chern numbers of such manifolds still satisfy the samenecessary relations and one might ask whether these are sufficient in order to guarantee theexistence of a connected realisation. In fact it turns out they are, as shown in [7].

Proposition 4.20. Let k denote the cardinality of the set of all partitions of n. If a givenk-tuple of integer numbers is realised by a disjoint union M1 t . . .tMr of 2n-dimensionalalmost complex manifolds, then the same k-tuple is realised by the connected almost complexmanifold

M1# . . .#Mr#(r−1)S2×S2n−2.

Remark. The main issue is of course to show that such a connected sum admits an almostcomplex structure, see [7].

4.2.2 Geography of symplectic manifolds.

Since symplectic manifolds admit in particular a compatible almost complex structure, theirChern numbers must also satisfy the congruence relations which necessarily hold in the al-most complex case. Let π(n) denote once more the cardinality of the set of all partitions of nand consider congruence relations among the Chern numbers of an almost complex manifoldof dimension 2n.

Definition 4.21. We call admissible those systems of π(n) integer numbers which satisfythe above congruence relations and ask ourselves which admissible π(n)-tuples may in factbe realised by a connected symplectic manifold of dimension 2n. We refer to the problem ofdetermining admissible π(n)-tuples which admit a connected symplectic realisation as to thesymplectic geography problem.

We list some results about symplectic geography in low dimension. In dimension 2, theChern number c1 coincides with the Euler number and satisfies c1 ≡ 0 (mod 2). Moreover,connectedness implies c1 ≤ 2. Any integer number satisfying the above relations can berealised by considering a compact Riemann surface of suitable genus.

In dimension 4, one has the necessary relation c21 + c2 ≡ 0 (mod 12). Any admissible

pair (p,q) satisfying the inequality p ≤ 2q admits a closed, connected, symplectic (in fact,even Kahler) realisation. More precisely, admissible pairs with p = 2q may be realised bya product of compact Riemann surfaces of suitable genus. Any admissible pair (p,q) withp < 2q, that is, below the line p = 2q, can be obtained by blowing up such a product a suitable


number of times. This is possible because blow-up at one point adds (−1,+1) to the pair ofChern numbers of the original manifold. The picture below should explain the use of theword geography.

q

p=2qp

Figure 4.1: Geography in dimension 4: the admissible pairs correspond to points of intersec-tion of the Z2-lattice with the lines p = −q + 12K. The arrow illustrates the effect of a onepoint blow-up.

In general, though, the problem of the geography of of symplectic 4-dimensional mani-folds is still open. Several authors have addressed in particular the problem of geographyof simply connected minimal symplectic 4-manifolds, that is, simply connected symplectic4-manifolds which do not contain symplectically embedded spheres with square −1. In thissituation one can show an interesting feature, namely that many pairs (p,q) may be realisedby such surfaces which do not admit a complex simply connected minimal realisation. Formore details, see [26].

There are also pairs which certainly do not admit a simply connected symplectic realisa-tion. Suppose for example that M is a simply connected symplectic 4-manifold. We want toshow that the sum of the Chern numbers of M cannot be negative.

Proposition 4.22. Suppose M is a closed, connected symplectic 4-manifold and that c21[M]+

c2[M] < 0. Then M cannot be simply connected.

Proof. Suppose we have c21[M]+ c2[M] < 0. One can make M minimal by blowing down a

finite number of disjoint exceptional spheres. Denote by M the minimal manifold resultingfrom the blow-down of say n such spheres. Then c2

1[M] = c21[M] + n, c2[M] = c2[M]− n

and M is still simply connected. Since any simply connected, closed, symplectic 4-manifoldmust satisfy c2 > 2, and c2

1[M] + c2[M] = c21[M] + c2[M] < 0, we see that we must have

c21[M] < c2

1[M]+ c2[M]−2 < −2. By a theorem of Liu (cf. [20], Theorem 13.39), M is thena ruled surface over a base of genus greater than 1, hence it cannot be simply connected.

Finally, in dimension six, one has the following system of relations:

c31 ≡ 0 (mod 2)

c1c2 ≡ 0 (mod 24)

c3 ≡ 0 (mod 2).

4.3. THE EIGHT-DIMENSIONAL CASE. 53

The standard form for an admissible triple is thus (2a,24b,2c). The symplectic geography inthis case is completely determined by the following theorem [13].

Proposition 4.23. Every admissible triple (2a,24b,2c) is realised as the system of Chernnumbers of a closed, connected, simply connected, symplectic six-manifold.

Having obtained a simply connected symplectic realisation for each admissible triple, onemight ask whether it is possible to find a realisation with arbitrary fundamental group. Wewill address this problem in Section 4.7.2.

4.3 The eight-dimensional case.

4.3.1 Congruence relations in dimension eight.

The group ΩU8 is generated by the equivalence classes of K4, K1 ×K3, K2

2 = K2 ×K2, K21 ×

K2 = K1 ×K1 ×K2, K41 = K1 ×K1 ×K1 ×K1 (compare (4.1)). The corresponding Chern

numbers are shown in Table 4.1.

c4 c1c3 c22 c2

1c2 c41

K4 5 50 100 250 625K1 ×K3 20 116 192 416 896

K22 9 54 99 216 486

K21 ×K2 12 60 96 204 432

K41 16 64 96 192 384

Table 4.1: Chern numbers of the generators of the group ΩU8 .

Because of the identification of ΩU8 with Γ4, any almost complex manifold M of dimen-

sion 8 must be C-equivalent to a disjoint union of a suitable number of copies of the abovegenerators and their inverses. This means that we are able to express the Chern numbers ofM as linear combinations with integer coefficients of the Chern numbers of the generators.More precisely, suppose

M ∼C x1K4 t x2K1K3 t x3K22 t x4K2

1 K2 t x5K41 , (4.5)

where xK denotes the disjoint union of x copies of K if x is positive of and the disjoint unionof |x| copies of the inverse of K if x is negative. Then the Chern numbers of M are given by:

c4[M] = 5x1 +20x2 +9x3 +12x4 +16x5

c1c3[M] = 50x1 +116x2 +54x3 +60x4 +64x5

c22[M] = 100x1 +192x2 +99x3 +96x4 +96x5

c21c2[M] = 250x1 +416x2 +216x3 +204x4 +192x5

c41[M] = 625x1 +896x2 +486x3 +432x4 +384x5.


Then if we compute the expressions

−c4[M]+ c1c3[M]+3c22[M]+4c2c2

1[M]− c41[M]

= 720x1 +1440x2 +720x3 +720x4 +720x5

2c41[M]+ c2

1c2[M] = 1500x1 +2208x2 +1188x3 +1068x4 +960x5

= 12 ·125x1 +12 ·184x2 +12 ·99x3 +12 ·89x4 +12 ·80x5

−2c4[M]+ c1c3[M] = 40x1 +76x2 +36x3 +36x4 +32x5

= 4 ·10x1 +4 ·19x2 +4 ·9x3 +4 ·9x4 +4 ·8x5

we immediately see that the Chern numbers of any almost complex 8-dimensional manifoldnecessarily satisfy the following congruence relations:

−c4 + c1c3 +3c22 +4c2c2

1 − c41 ≡ 0 (mod 720)

2c41 + c2

1c2 ≡ 0 (mod 12) (4.6)

−2c4 + c1c3 ≡ 0 (mod 4).

We claim that these relations are also sufficient. Suppose in fact that we are given a quin-tuple of integer numbers (c4,c1c3,c2

2,c21c2,c4

1) satisfying them. Then there exist integers(a, j,k,m,b) such that

a = c4

720 j = −c4 + c1c3 +3c22 +4c2c2

1 − c41

12k = 2c41 + c2

1c2 (4.7)

4m = −2c4 + c1c3

b = c41

and the above system is equivalent to

c4 = a

c1c3 = 4m+2a

c41 = b

c21c2 = 12k−2b

3c22 = 720 j−a−4m−48k +9b.

From this we see that there is a one-to-one correspondence between quintuples of integerssatisfying (4.6) and quintuples (a,b, j,k,m) subject to the condition a+m ≡ 0 (mod 3).

In Table 4.2 we collect the values of the parameters (a,b, j,k,m) for the group generatorsin dimension 8.

In order to prove our claim that the relations (4.6) represent a sufficient condition for aquintuple of integer numbers to appear as the system of Chern numbers of an almost complexmanifold of dimension 8, we have to show that given a quintuple (a,b, j,k,m) satisfying thecondition a+m≡ 0 (mod 3) there exist integer coefficients x1, . . . ,x5 such that the parameters


a m j k b

K4 5 10 1 125 625K1 ×K3 20 19 2 184 896

K22 9 9 1 99 486

K21 ×K2 12 9 1 89 432

K41 16 8 1 80 384

Table 4.2: Parameters for the generators of the group ΩU8 .

of the manifold x1K4tx2K1K3tx3K22 tx4K2

1 K2tx5K41 coincide with the given quintuple. For

this to happen, in turn, the system

a = 5x1 +20x2 +9x3 +12x4 +16x5

m = 10x1 +19x2 +9x3 +9x4 +8x5

j = x1 +2x2 + x3 + x4 + x5

k = 125x1 +184x2 +99x3 +89x4 +80x5

b = 625x1 +896x2 +486x3 +432x4 +384x5

has to admit an integer solution. This is indeed the case, namely the system admits a solution

x1 =23

a+96 j +83

m+3b−16k

x2 =83

a+600 j +413

m+17b−91k

x3 = −a−80 j−4m−3b+16k

x4 = −253

a−1920 j− 1243

m−54b+289k

x5 =103

a+705 j +463

m+20b−107k

and the condition a+m ≡ 0 (mod 3) guarantees that the xi’s are in fact integer.

We would like to show how the relations can also be written down starting from theRiemann-Roch theorem. In dimension 8,

ex j −1 = x j +x2

j

2+

x3j

3!+

x4j

4!.


Then comparing with [22, p.188], we get for instance

e1 =4

∑j=1

(ex j −1)

= ∑x j +12 ∑x2

j +13! ∑x3

j +14!

x4j

= s1 +12

s2 +13!

s3 +14!

s4

= c1 −12

(c21 −2c2)+

16

(c31 −3c1c2 +3c3)

− 124

(c41 −4c2

1c2 +2c22 +4c1c3 −4c4).

Similarly, the other classes ei are given by

e2 = c2 −12(c1c2 − c3)+

112

(14c4−8c1c3 − c22 +2c2

1c2)

e3 = c3 −12

(c1c3 −4c4)

e4 = c4.

Notice that the components of ei have all dimension greater than or equal to 2i. Congruencerelations for the Chern numbers of M arise from integrality of the numbers

〈T4, [M]〉 =1

720(−c4 + c1c3 +3c2

2 +4c21c2 − c4

1)

〈(e1 ·T )4, [M]〉 =1

12c1c3 +

16

c4

〈(e21 ·T )4, [M]〉 =

112

c21c2 +

16

c41

〈(e31 ·T )4, [M]〉 =

14

(−c41 −6c2

1c2)

〈(e2 ·T )4, [M]〉 =1

12(c1c3 +14c4)

〈(e22 ·T )4, [M]〉 = c2

2

〈(e3 ·T )4, [M]〉 = 2c4

〈(e4 ·T )4, [M]〉 = c4.

This leads to the following system of congruence relations

−c4 + c1c3 +3c22 +4c2c2

1 − c41 ≡ 0 (mod 720)

2c41 + c2

1c2 ≡ 0 (mod 12)

2c4 + c1c3 ≡ 0 (mod 12),

which is easily seen to be equivalent to (4.6).


4.3.2 The symplectic case.

We now turn our attention to the geography of symplectic 8-dimensional manifolds. We willkeep working with the parameters (a, j,k,m,b) rather than with the Chern numbers them-selves. The result which we want to prove can be summarised by saying that all quintuples(a,b, j,k,m) satisfying the condition a+m ≡ 0 (mod 3) admit a symplectic realisation.

Theorem 4.24. Given a quintuple of integer numbers (a, j,k,m,b), subject to the additionalcondition a + m ≡ 0 (mod 3), there exists a closed, connected, symplectic 8-dimensionalmanifold M such that the given parameters are related to the Chern numbers of M by thesystem of equations (4.7).

In view of the correspondence between quintuples (a, j,k,m,b) satisfying the conditiona+m ≡ 0 (mod 3) and admissible quintuples, the Theorem immediately implies that the con-gruence relations (4.6) are not only necessary, but also sufficient for a given quintuple of in-teger numbers to occur as the system of Chern numbers of a closed connected 8-dimensionalsymplectic manifold.

Corollary 4.25. Given a quintuple of integer numbers (n1,n2,n3,n4,n5) which is admissible,i.e., which satisfies the system of congruence relations

−n1 +n2 +3n3 +4n4−n5 ≡ 0 (mod 720)

2n5 +n4 ≡ 0 (mod 12)

−2n1 +n2 ≡ 0 (mod 4)

there exists a closed, connected, symplectic 8-dimensional manifold M realising the givenquintuple as its system of Chern numbers, namely (c4[M], c1c3[M], c2

2[M], c21c2[M], c4

1[M])=(n1,n2,n3,n4,n5).

4.3.3 Behaviour of the parameters under blow-up.

From the blow up formulae (A.3), (A.5), (A.6) for the Chern numbers, we obtain the follow-ing expressions for the transformation of the parameters of an eight dimensional manifold Munder blow-up.

• Blow-up at a point:

a′ = a+3

4m′ = 4m

720 j′ = 720 j (4.8)

12k′ = 12k−180

b′ = b−81

• Blow-up along a symplectically embedded curve C of genus g and with normal bundle


νC:

a′ = a+4(1−g)

4m′ = 4m−4(1−g)

720 j′ = 720 j (4.9)

12k′ = 12k−144(1−g)−36〈c1(νC), [C]〉b′ = b−64(1−g)−16〈c1(νC), [C]〉

• Blow up along a symplectic four-dimensional submanifold X , with normal bundle νX :

a′ = a+ c2[X ]

4m′ = 4m+ c21[X ]−3c2[X ]

720 j′ = 720 j (4.10)

12k′ = 12k−13c21[X ]− c2[X ]−18〈c1(X)c1(νX ), [X ]〉

−6〈c21(νX ), [X ]〉

b′ = b−6c21[X ]−8〈c1(X)c1(νX ), [X ]〉−3〈c2

1(νX ), [X ]〉+〈c2(νX ), [X ]〉

With reference to the proof of Proposition 4.24, notice that the parameter j defined in(4.7) is invariant under blow up.

Following the strategy of Halic [13], we will start by producing examples of closed sym-plectic manifolds realising any given value of j and admitting enough symplectic submani-folds. Then we will show that by blowing up along these submanifolds a suitable number oftimes, the other parameters may also be varied so as to obtain any prescribed quintuple.

4.4 Building blocks.

4.4.1 Elliptic surfaces.

Definition 4.26. An elliptic surface is by definition a complex surface together with a holo-morphic map to a complex curve C whose fibres are tori, except for a finite number of so-called singular fibres.

By abuse of terminology, such a map is also called a fibration and we are particularlyinterested in the case where the base of the fibration is the complex projective line CP1.Consider for example two distinct nondegenerate cubic curves C0 and C1 in CP2, given asthe zero sets of some homogeneous cubic polynomials Pi, i = 0,1. We may assume that C0

and C1 intersect transversely in exactly nine positive points q1, . . . ,q9. For any other pointp ∈CP2−q1, . . . ,q9, there exists a unique µ = [w0 : w1]∈CP1 such that p lies on the cubiccurve

Cµ = [z0 : z1 : z2] ∈ CP2 |(w0P0 +w1P1)(z0,z1,z2) = 0.

4.4. BUILDING BLOCKS. 59

In fact, if p = [x : y : z], we may assume that C0(x,y,z) 6= 0 and let

w0 = −C1(x,y,z)C0(x,y,z)

w1.

Then obviously (w0C0 +w1C1)(x,y,z) = 0. In this way one gets a well defined map

f : CP2 −q1, . . . ,q9 −→ CP1,

which maps a point p to the element [w0 : w1]∈CP1 satisfying the condition that p belongs tothe curve defined by w0C0 +w1C1. The map f extends, after blowing up the nine intersectionpoints of C0 and C1, to a map

f : CP2#9CP2 −→ CP1

such that the fibre f−1(µ) =Cµ, µ∈CP1, is a cubic curve. Each exceptional sphere is a sectionof f . All but a finite number of fibres are smooth and have genus g = 1

2(d − 1)(d − 2) =1, that is, except for a finite number of singular fibres, they are topologically tori. Thatthere need to be singular fibres as well can be seen from the fact that c2(CP2#9CP2) isdifferent from zero. The types of singular fibres depend on the choice of polynomials P0andP1. The manifold CP2#9CP2 admits a symplectic (in fact, even Kahler) form, namely theone obtained by blow-up from the standard Kahler form on CP2. The regular fibres of f arecomplex submanifolds, hence again Kahler, hence in particular symplectic submanifolds. Aneighbourhood of a regular fibre coincides with a neighbourhood of the 9-times blow-up ofa smooth cubic curve in CP2: therefore we see that these regular fibres have trivial normalbundle.

From now on we will denote by E(1) precisely the manifold CP2#9CP2 , equippedwith such an elliptic fibration. We assume that this fibration has only cusp fibres as singularfibres (these are fibres obtained by blowing up curves ambiently isotopic to [z0 : z1 : z2] ∈CP2 |z2z2

1 = z30 = 0).

The homology of E(1) is the homology of a connected sum and it is given by

H2(E(1)) ∼= H2(CP2#9CP2) ∼= H2(CP2;Z)⊕9H2(CP2;Z)

Let h be the generator of H2(CP2;Z), ei the exceptional sphere of the i-th blow up, i =1, . . . ,9. Then 〈h,e1, . . . ,e9〉 is a basis for H2(E(1)) with intersection matrix 〈1〉⊕9〈−1〉.

A regular fibre represents the class f = 3h−∑9i=1 ei: another basis for H2(E(1)) is given

by 〈 f ,e9,e1 − e2, . . . ,e7 − e8,−h + e6 + e7 + e8〉 and the intersection matrix with respect tothis basis (cf. Section 1.5) is

[0 11 −1

]⊕ (−E8).

The first Chern class of E(1) is PD(3h−∑ei) (cf. Corollary 3.22 and following Remark),which gives for the Chern numbers the values c2

1[E(1)] = 0 and c2[E(1)] = 12.Let F be a regular fibre in E(1): we can perform the symplectic sum of two copies

of E(1) along two such fibres: the resulting manifold E(2) := E(1)#FE(1) still fibres over


CP1 (regarded as the connected sum of two copies of itself) with fibres which are tori, i.e.,it is again an elliptic surface. The second homology group H2(E(2)) admits a basis withintersection matrix

2(−E8)⊕3

[0 11 −2

]

where the elements of square 0 are tori and the other spheres (cf. [8, p. 72]). By the expressionfor the first Chern class of a symplectic sum along tori with trivial normal bundle, (3.16),c1(E(2)) vanishes. In fact

c1(E(1)#FE(1)) = c1(E(1))+ c1(E(1))−2PD[F] = 2 f −2 f = 0.

Inductively, one can define E(n+1) := E(n)#FE(1) = #n+1F E(1) and find a basis for its

second homology with corresponding intersection matrix

n(−E8)⊕2(n−1)

[0 11 −2

]⊕

[0 11 −n

](4.11)

We denote the elements of this basis by

〈τi j , i = 1, . . . ,n; j = 1, . . . ,8; αk,βk,k = 1, . . . ,2(n−1); f ,σ〉,

where σ denotes the class of a section of E(n), which is obtained by pasting together nsections of E(1).

The first Chern class is c1(E(n)) = (2−n) f , so the Chern numbers are given by

c21[E(n)] = 0

c2[E(n)] = 12n.

Definition 4.27. The nucleus N(n) of the elliptic surface E(n) consists of a neighbourhoodof the union of a singular fibre and a section of the fibration.

If we consider the nucleus of the elliptic surface E(n), we have that H2(N(n)) ∼= Z2 and

the corresponding intersection matrix is the last summand in QE(n), namely

[0 11 −n

].

4.4.2 Other building blocks.

Other “building blocks” are obtained as follows [9]:

• In T 2 ×T 2 consider the union of the two tori T 2 ×p∪p×T2. The intersectionpoint is positive and transverse: in a neighbourhood of this point, the union of the twotori looks like the set z1z2 = 0 near the origin of C2, that is, like two 2-dimensionaldiscs intersecting in one point of the 4-dimensional ball. We may replace it then withthe annulus z1z2 = ε, performing the gluing so that the boundary of the annuluscomes to coincide with that of the two discs. This eliminates the intersection point,without changing the ambient manifold. The process increases the Euler number by2, and the resulting submanifold is still symplectically embedded in T 2 × T 2, in the

4.5. CONSTRUCTION OF THE EXAMPLES. 61

same homology class as T 2 ×p∪p×T 2. After this, we blow up twice to obtaina symplectic genus 2 surface F2 with square 0 in Q = T 4#2CP2 . Then Q contains asymplectic torus F , disjoint from F2. This F is obtained from

F ′ = (x1,x2,x3,x4) ∈ T4 = R4/Z4 |x2 = x4 = 0,

which is in fact Lagrangian, by perturbing the symplectic form. Then F will be disjointfrom F2 provided p = (0,c) with c 6= 0.

• Let p and q be distinct points in T 2 and consider the two tori T 2× p and q×T 2 in T 4 =T 2 ×T 2. Blow up the intersection point (q, p) to obtain two disjoint tori with square−1, then blow up 16 more times to reduce the square of both to −9. Now take thesymplectic sum of the resulting manifold T 4#17CP2 with 2 copies of CP2 along cubiccurves: denote the final result of these operations by S. Then S is a simply connected,symplectic 4-manifold, containing disjoint symplectically embedded surfaces of genus1 and 2 with trivial normal bundle.

• Consider a curve of degree 4 with one transverse double point in CP2. The genus ofsuch a curve is given by g = (d−1)(d−2)

2 −#(nodes) and in our case this number is 2. Wecan get a smooth surface by blowing up the the double point: this surface representsthe homology class 4h−2e, hence it has square 12. We thus need to blow up 12 moretimes to get a smooth submanifold with genus 2 and square 0 in CP2#13CP2. Finallyblow up three extra points, away from F2, to get the manifold P ∼= CP2#16CP2. P is asymplectic simply connected manifold with Chern numbers c2

1 = −7 and c2 = 19.

4.5 Construction of the examples.

4.5.1 Symplectic sphere bundles, Part II.

Let (N,β) be a closed symplectic 4-dimensional manifold, for example, one of the abovebuilding blocks, and E→N a complex line bundle over N. We take a step back and consideronce more the bundle ρ : S→N with fibre S2 over N, obtained by projectifying the complexrank two bundle E ⊕C over N.

Then (3.4) gives us the expression for the Chern classes of S: from this and the ringstructure (3.1) we can compute the corresponding Chern numbers, which are given by

c31[S] = 6c2

1[N]+2〈c21(E), [N]〉

c1c2[S] = 2(c21[N]+ c2[N])

c3[S] = 2c2[N].

The examples we will consider are 8-dimensional symplectic manifolds of the form M =S×F , with F a compact Riemann surface of genus g. Using a product formula we can easily


compute the Chern numbers of M.

c4[M] = 2(1−g)c3[S] = 4(1−g)c2[N]

c1c3[M] = 2(1−g)(c1c2[S]+ c3[S]) = 4(1−g)(c21[N]+2c2[N])

c22[M] = 4(1−g)c1c2[S] = 8(1−g)(c2

1[N]+ c2[N])

c21c2[M] = 2(1−g)(c3

1[S]+2c1c2[S])

= 4(1−g)(5c21[N]+2c2[N]+ 〈c2

1(E), [N]〉)c4

1[M] = 8(1−g)c31[S] = 16(1−g)(3c2

1[N]+ 〈c21(E), [N]〉).

The next step will be to consider symplectic submanifolds of M which are of the formB×pt, with B a symplectic submanifold of S and pt ∈ F . For such submanifolds, thenormal bundle in M coincides with the Whitney sum of the normal bundle in S and a copyof the trivial line bundle, which we denote by C. This implies in particular an equivalence ofChern classes

c(νMB) = c(νSB⊕C) = c(νSB).

We consider for example sections N+ and N− of S, corresponding to the embeddings ofN in S = E0 ∪∂E0 E0 (cf. Lemma 2.19) as the zero section of E and E, respectively. In thiscase, the characteristic numbers which appear in the blow-up formulae are given by

c21[N±] = c2

1[N]

c2[N±] = c2[N]

〈c21(νMN±), [N±]〉 = 〈c2

1(E), [N]〉〈c1(νMN±)c1(N±), [N±]〉 = ±〈c1(N)c1(E), [E]〉.

Let s be any such section and assume that F is a symplectically embedded curve in N:then it lifts along s to a symplectically embedded curve in S. Moreover, the square of F willchange by an amount equal to the product 〈c1(E), [F ]〉. By stretching the terminology, wecall here square of F also the number resulting from evaluating the first Chern class of thenormal bundle of F (or rather, s(F)) in S on its fundamental homology class. More preciselywe have:

Lemma 4.28. In the situation described above, the square of the lift of an embedded curveF is given by

〈c1(ν(s(F),S)), [s(F)]〉 = 〈c1(ν(F,N)), [F ]〉+ 〈c1(E), [F ]N〉,

where ν(·, ·) denotes the normal bundle of an embedding and s is the section under consider-ation.

Proof. We refer to the following commutative diagram for the notation:

s(F)j−−−−→ S

∼=y

yρ

Fi−−−−→ N

4.5. CONSTRUCTION OF THE EXAMPLES. 63

We have an isomorphism of vector bundles:

ν(s(F),S) ∼= ν(s(F),s(N))⊕ν(s(N),S)|s(F)

∼= ρ∗ν(F,N)⊕ρ∗E|s(F),

which implies a corresponding equivalence on the level of cohomology classes, namely:

c1(ν(s(F),S)) = ρ∗c1(ν(F,N))+ρ∗i∗c1(E).

We now evaluate on [s(F)] and get

〈c1(ν(s(F),S)), [s(F)]〉 = 〈ρ∗c1(ν(F,N)), [s(F)]〉+ 〈ρ∗i∗c1(E), [s(F)]〉= 〈ρ∗c1(ν(F,N)),s∗[F]〉+ 〈ρ∗i∗c1(E),s∗[F ]〉= 〈s∗ρ∗c1(ν(F,N)), [F ]〉+ 〈s∗ρ∗i∗c1(E), [F ]〉= 〈c1(ν(F,N)), [F ]〉+ 〈i∗c1(E), [F ]〉= 〈c1(ν(F,N)), [F ]〉+ 〈c1(E), [F ]N〉.

Notice that, in particular, if c1(E)∩ [F ] = 0, then the lifted curve has the same square inS as the original one in N, i.e.,

〈c1(ν(s(F),S)), [s(F)]〉 = 〈c1(ν(F,N)), [F ]〉.

Recall that we may assume the symplectic form ωK = Kρ∗β+η on S to be integral andto represent the class Kρ∗[β]+ ξ, with [β] ∈ H2(N;Z) (cf. Lemma 3.11). As far as the lastcondition is concerned, we have already seen that it can always be realised without changingthe Chern classes of N. In fact, given any symplectic form ω′, we are able to find an integralsymplectic form ω, which is obtained by first approximating ω′ and then rescaling. Supposeω′ tames an almost complex structure J: since the taming condition is open and an invariantwith respect to rescaling, we see that J is also a tame almost complex strcuture for the integralform ω′.

This, together with the following lemma, implies that symplectically embedded curves(2-dimensional submanifolds) also remain symplectically embedded with respect to the newintegral symplectic form.

Lemma 4.29. A smooth 2-dimensional submanifold F of a symplectic 4-manifold (N,ω) issymplectically embedded if and only if it is a J-holomorphic curve with respect to some tamealmost complex structure J on (N,ω).

Proof. Suppose first that the inclusion i : (F, j)→(N,J) is a J-holomorphic map, that is, dij = J di. Let v ∈ TF . If i∗ω(v,w) = 0 for all w ∈ T F, then in particular

0 = i∗ω(v, jv) = ω(di(v),di j(v))

= ω(di(v),J di(v)),


but the latter is strictly positive by the taming condition unless v = 0. Hence i∗ω is nondege-nerate on TF , i.e., it is a symplectic form.

Conversely, suppose F is a symplectic submanifold. Then by Lemma 3.3 there exists atame almost complex structure JN on N such that its restriction to TF is again an almost com-plex structure: in fact, JN |T F must be homotopic to j, since there exists only one homotopyclass of almost complex structures on every orientable surface (SO(2) = U(1)). It is theneasy to perturb JN so that in fact JN |T F = j.

With an integral symplectic form at our disposal, we may apply Donaldson’s existencetheorem and obtain a whole family of symplectic submanifolds Xλ of S which, for suf-ficiently large λ ∈ Z, realise the Poincare dual of λ[ωK ], i.e., PDS[Xλ] = λ(ξ + K[π∗β]) inH2(S;Z).

Let i denote the inclusion Xλ in M. We have the relation

i∗c(S) = c(Xλ)∪ c(νSXλ)

and since c1(νSXλ) = e(νMXλ) = i∗PDM [Xλ], we can rewrite it as

i∗c(M) = c(Xλ)∪ (1+ i∗λ(ξ+K[π∗β])).

From this relation, using

〈i∗y, [Xλ]〉 = 〈y∪ (λξ+λK[π∗β]), [M]〉 for all y ∈ H4(M)

and the cohomology ring structure of M, we can compute the invariants of Xλ (see the Ap-pendix for explicit computations). They are:

c2[Xλ] = λc2[N]+λ(λ−1)〈c1(N)c1(L), [N]〉−2λK(λ−1)〈c1(N)[β], [N]〉+λ2(λ−1)〈c2

1(E), [N]〉+λ2K(2−3λ)〈c1(E)[β], [N]〉+λ2K2(3λ−2)〈[β]2, [N]〉

c21[Xλ] = λc2

1[N]+2λ(λ−1)〈c1(N)c1(L), [N]〉+4λK(1−λ)〈c1(N)[β], [N]〉+λ(λ2−2λ+1)〈c2

1(E), [N]〉 (4.12)

+λ2K(4−3λ)〈c1(E)[β], [N]〉+λ2K2(3λ−4)〈[β]2, [N]〉

〈c21(νMXλ), [Xλ]〉 = λ3〈c2

1(E), [N]〉−3λ3K〈c1(E)[β], [N]〉+3λ3K2〈[β]2, [N]〉

〈c1(Xλ)c1(ν), [Xλ]〉 = −λ2〈c1(N)c1(L), [N]〉+2λ2K〈c1(N)[β], [N]〉+λ2(1−λ)〈c2

1(E), [N]〉+λ2K(3λ−2)〈c1(E)[β], [N]〉+λ2K2(2−3λ)〈[β]2, [N]〉.

4.6. THE BLOW-UP SYSTEMS. 65

4.6 The blow-up systems.

Now we have introduced all the elements necessary for the proof of Theorem 4.24. The proofitself follows that of Halic for the case of dimension 6. For any given j, that is, we will showthat it is possible to construct a symplectic manifold realising j and with enough symplecticsubmanifolds so that we can vary the other parameters and eventually realise all admissiblequintuples. We construct our examples distinguishing three main cases.

4.6.1 Realising sets of parameters with j ≥ 1.

We start by considering examples of symplectic 8-dimensional manifolds for which the pa-rameter j is greater than or equal to 1. In order to produce such examples, we perform thesymplectic sum of the manifolds Q and E(n) of paragraph 4.4.2 along embedded tori withsquare zero. Before doing so, though, we blow up one extra point in Q, away from the torusalong which we intend to perform the sum.

The result of these operations is the manifold

Xn := Q∗#F E(n),

which is a simply connected symplectic manifold, with Chern numbers c21 = −3 and c2 =

3+12n.Following our discussion of the first Chern class of such a sum, we may write

c1(Xn) = c1(Q∗)+ c1(E(n))−2PD([F]).

Assume that the torus F ⊂ E(n) is actually contained in N(n), the nucleus of the ellipticsurface (cf. Definition 4.27), and recall that we denote by τi j, i = 1, . . . ,n, j = 1, . . . ,8, theelements of the basis of H2(E(n)) corresponding to the n copies of the (−E8)-block in theintersection matrix (4.11). The τi j’s are represented by submanifolds of the complement ofN(n), hence disjoint from F : for this reason they represent homology classes in Xn (whichwe still denote by τi j) and we may consider their Poincare duals, which will be elements ofH2(Xn).

According to this interpretation of the elements τi j ∈ H2(Xn) we consider the complexline bundle L over Xn, specified by its first Chern class

c1(L) =n−1

∑i=1

2PD(τi1)+3

∑j=1

PD(τn j)+PD(τn8) ∈ H2(Xn;Z),

and compute 〈c2

1(L), [Xn]〉 = −8(n−1)−12〈c1(L)c1(Xn), [Xn]〉 = 0.

Now consider the manifold S = P(L⊕C). Let s : Xn→S be the section which embeds Xn

in S as P(L⊕0) = (Xn)−. Denote by E the exceptional sphere of the last blow-up in Q∗

and by e = [E] its fundamental homology class. Similarly, if E− = s(E) is the lift of E along


the section s, let e− = [E−] = [s(E)] be the corresponding homology class, so that we havethe following commmutative diagram:

E− −−−−→ S

ρy∼=

yρ

E −−−−→ Xn

Then 〈c1(L|E),e〉 = PD−1c1(L) · e = 0, so by Lemma 4.28 we have

〈c1(νME−),e−〉 = e2 + 〈c1(L|E),e〉 = e2.

Observe that if F2 denotes the symplectically embedded surface of genus 2 in Q∗, it is disjointfrom any representative of the classes [τi j] and therefore it also lifts to F2− ⊂ Xn− with

〈c1(νMF2−), [F2]−〉 = 0.

Finally recall that S admits an integral symplectic form ωK and hence, for λ large enough,symplectic submanifolds Xλ, realising the Poincare dual of λ[ωK ], whose invariants are givenby the expressions in (4.12).

Take the product of S with S2 to obtain the simply connected symplectic 8-dimensionalmanifold M = S× S2. The parameter j (recall: j was defined by the congruence relation−c4 + c1c3 + c2

2 + 4c2c21 − c4

1 = 720 j) in this case takes on precisely the value n. The otherparameters are

a = 48n+12

4m = −12

12k = −192n−468

b = −128n−208.

Now blow up M at x points, y copies of E, z copies of F2, u copies of Xn− and vcopies of Xλ. Denote by M the manifold obtained after performing these blow-ups and let(a′,m′, j′,k′,b′) be the parameters associated to M. Then j′ = j (recall that we already re-marked that j is invariant under blow-up), whereas by applying the blow-up formulae (4.8),(4.9) and (4.10) we find the following expressions for the other parameters:

a′ = 48n+12+3x+4y−4z+(12n+3)u+b1v

4m′ = −12−4y+4z+(−36n−12)u+b2v (4.13)

12k′ = −192n−468−180x−108y+144z+(36n+60)u+b3v

b′ = −128n−208−81x−48y+64z+(24n+30)u+b4v.

The coefficients coming from blowing up along the submanifold Xλ, whose invariants are


computed in the Appendix, are given by:

b1 = λc2(N)+(λ3 −λ2)c21(E)+(3λ3K2 −2λ2K2) [β]2

+(λ2 −λ)c1(N)∪ c1(E)+(2λK−2λ2K)c1(N)∪ [β]

+(2λ2K −3λ3K)c1(E)∪ [β]

b2 = λc21(N)−3λc2(N)+(λ2−2λ3 +λ)c2

1(E)+(2λ2−6λ3K2) [β]2

+(λ−λ2)c1(N)∪ c1(E)+2(λ2K −λK)c1(N)∪ [β]

+(6λ3K −2λ2K)c1(E)∪ [β]

b3 = −13λc21(N)−λc2(N)+(9λ2−2λ3−13λ)c2

1(E)

+(18λ2K2 −6λ3K2) [β]2 +(27λ−9λ2)c1(N)∪ c1(E)

+(18λ2K −54λK)c1(N)∪ [β]+ (6λ3K −18λ2K)c1(E)∪ [β]

b4 = −6λc21(N)+(4λ2−6λ−λ3)c2

1(E)+(8λ2K2 −3λ3K2) [β]2 +

(12λ−4λ2)c1(N)∪ c1(E)+(8λ2K −24λK)c1(N)∪ [β]

+(3λ3K −8λ2K)c1(E)∪ [β],

where we have suppressed evaluation on the fundamental class of [N] from the notation.We regard (4.13) as a linear system in the variables x, y, z, u, v. If we can prove that

for arbitrary parameters a′, m′, k′, b′, satisfying the additional condition a′ +m′ ≡ 0(mod 3),the system admits a quintuple of positive, integer solutions, then we will have shown thatwe can realise all such parameters, precisely by performing on M the sequence of blow-upscorresponding to the solutions (x,y,z,u,v).

The solutions of system (4.13) are:

x = [(32n+13)λ3K2[β]2 + r1(λ,K)]v+(8n+

103

)a′ +

(32n+

403

)m′ +(−128n−48)k′+(24n+9)b′+

640n2 +224n

y =[(

12n+92

)λ3K2[β]2 + r2(λ,K)

]v+(3n+3)a′+

(12n+8)m′+(−48n−21)k′+(9n+4)b′+

240n2 +32n+1

z = [(48n+18)λ3K2[β]2 + r3(λ,K)]v+(12n+6)a′+

(48n+21)m′+(−192n−69)k′+(36n+13)b′+

960n2 +272n+4

u = [4λ3K2[β]2 + r4(λ,K)]v+a′+4m′−16k′+

3b′+80n.

First of all notice that these solutions are, indeed, integer, because of the additional con-dition a′ + m′ ≡ 0 (mod 3) and our freedom of choice of the variable v. Moreover, one canobserve that the ri(λ,K) are polynomials of degree at most 2 in λ and 2 in K, with coefficients


depending on c21(N),c2(N),c2

1(E), [β]2,c1(N)∪ c1(E),c1(N)∪ [β],c1(E)∪ [β], evaluated on[N]; recall also that 〈[β]2, [N]〉 is strictly positive, because β2 = β∧β is a volume form on N:by the previous two remarks we conclude that, by choosing λ large enough, we may ensurethe positivity of the v-coefficients and consequently, again by a choice of v sufficiently large,positivity of all variables.

4.6.2 The case j = 0.

In order to show that all quintuples of parameters with j = 0 admit a symplectic realisation,we start again by constructing a manifold with j = 0. We consider the 4-manifold Q∗ and thecomplex line bundle L defined by c1(L) = −2e3, with e3 denoting the exceptional divisor ofthe last blow-up. Since

c1(Q∗) = c1(T

4)−3

∑i=1

ei,

we see that 〈c2

1(L), [Q∗]〉 = −4〈c1(L)c1(Xn), [Q∗]〉 = −2.

We proceed to construct S = P(L⊕C) and M = S× S2 as in the previous section. ThenM realises j = 0, as required, and the other parameters are

a = 12

4m = −24

12k = −488

b = −218.

We let Q∗− be the image of the embedding of Q∗ in M as P(L⊕0). Then Q∗

− contains asphere E with square −1 (the exceptional sphere of either the first or the second blow-up) anda genus 2 surface F2. These curves intersect in S, but since the cup product of their Poincareduals and c1(L) vanishes, by Lemma 4.28 they provide disjoint submanifolds E ×pt andF2 ×pt of M = S×S2 with the same genus and square.

Together with the submanifold Q∗−, we consider as in the previous cases submanifolds

Xλ, realising the Poincare duals of multiples λωK of some integral symplectic form ωK onM.

We are now able to write down the blow-up system for M, where we blow up at x points,y copies of E, z copies of F2, u copies of Q∗

− and v copies of Xλ. Then the parameters of our8-dimensional manifold transform according to the following expressions

a′ = a+3x+4y−4z+3u+b1v

4m′ = 4m−4y+4z−12u+b2v (4.14)

12k′ = 12k−180x−108y+144z+24u+b3v

b′ = b−81x−48y+64z+14u+b4v,


where the bi’s are once again the coefficients corresponding to blow-up along submanifoldsbelonging to the family Xλ.

The solutions in this case are

x = [(13λ3K2[β]2 + r1(λ,K)]v+103

a′ +403

m′−48k′+9b′

y =[17

2λ3K2[β]2 + r2(λ,K)

]v+4a′+12m′−37k′+7b′+1

z = [22λ3K2[β]2 + r3(λ,K)]v+7a′+25m′−85k′+16b′+4

u = [4λ3K2[β]2 + r4(λ,K)]v+a′+4m′−16k′+3b′.

Once again, we may observe that all solutions are integer and that by a suitable choice ofλ and v we may assume them to be positive, as well. Hence all admissible quintuples withj = 0 admit a symplectic realisation.

4.6.3 Negative values of j.

We are left with only the case j < 0 to take care of. For this we construct a 6-dimensionalmanifold S as in the cases of positive values of the parameter j and then define M to be theproduct of S with a compact Riemann surface of genus two. Notice that in this case therealisation will not be simply connected.

The only difference in the blow-up system occurs in the parameters corresponding to themanifold Σ which is blown up: these have in fact opposite sign. Therefore the blow-up systemhas the form

a′ = −48n−12+3x+4y−4z+(12n+3)u+b1v

4m′ = 12−4y+4z+(−36n−12)u+b2v

12k′ = 192n+468−180x−108y+144z+(36n+60)u+b3v

b′ = 128n+208−81x−48y+64z+(24n+30)u+b4v

and the solutions are given by

x = [(32n+13)λ3K2[β]2 + r1(λ,K)]v+(8n+103

)a′ +

(32n+403

)m′ +(−128n−48)k′+(24n+9)b′+

−640n2−224n


y = [(12n+92)λ3K2[β]2 + r2(λ,K)]v+(3n+3)a′+

(12n+8)m′+(−48n−21)k′+(9n+4)b′+

−240n2−32n−1

z = [(48n+18)λ3K2[β]2 + r3(λ,K)]v+(12n+6)a′+

(48n+21)m′+(−192n−69)k′+(36n+13)b′+

−960n2−272n−4

u = [4λ3K2[β]2 + r4(λ,K)]v+a′+4m′−16k′+

3b′−80n.

The same considerations as to positivity and integrality apply as in the case of positive j.

4.7 Some final remarks.

4.7.1 Kahler manifolds.

We have shown that, in dimension 8, the geography of symplectic manifolds coincides withthat of almost complex manifolds. It is then natural to ask whether this is true also for thegeography of Kahler manifolds. Our work unfortunately does not provide a positive answerto this question. The examples we construct, in fact, cease to be Kahler at the point wherethey are blown up along submanifolds of Donaldson’ s type. This is because the latter are notnecessarily complex submanifolds.

4.7.2 Geography with fundamental group.

We would like to conclude by briefly addressing the question of geography with fundamentalgroup, that is: to which extent is it possible to prescribe Chern numbers and fundamentalgroup of a symplectic manifold at the same time.

Observe that Halic’ result yields in dimension 6 simply connected realisations for alladmissible triples. In dimension 4, on the other hand, if (p,q) is an admissible pair withp + q < 0, there exists no simply connected symplectic manifold with (c2

1,c2) = (p,q), aswas shown in Proposition 4.22.

If G is any finitely presentable group, Gompf has shown that there exists a closed sym-plectic 4-manifold with fundamental group G. This manifold, moreover, may be assumed tosatisfy certain additional properties.

Theorem 4.30 (Gompf). Let G be any finitely presentable group. Then there is a closedsymplectic 4-manifold MG with π1(MG) ∼= G. Furthermore we may assume:

(i) c21[MG] = 0, c2[MG] = 12r > 0;

(ii) MG contains a symplectic torus T with square 0 and inclusion i : T→MG inducing thetrivial map on π1.

4.7. SOME FINAL REMARKS. 71

The proof can be found in [9] and relies in fact on the symplectic connected sum con-struction.

Remark. The existence of a four-dimensional symplectic manifold MG with π1(MG) ∼= Gfor every finitely presentable group G is another feature that distinguishes symplectic fromKahler manifolds. The abelianisation of the fundamental group of a Kahler manifold, in fact,necessarily has even rank, since (π1)ab = π1/[π1,π1] ∼= H1 (cf. Example 2.4).

Theorem 4.30 can be applied to improve partially Halic’ result in dimension 6 and showthat some admissible triples may be realised by a closed connected symplectic manifold Mhaving a given finitely presentable fundamental group G.

Proposition 4.31. For every admissible triple (2a,24b,2c) with b ≤ −2 and every finitelypresentable group G there exists a closed symplectic 6-manifold M such that

c31[M] = 2a

c1c2[M] = 24b

c3[M] = 2c

π1(M) ∼= G.

Proof. Let S be as in Section 4.4.2 and denote by X the symplectic connected sum of S withE∗

n , the blow-up at one point of the elliptic surface En. In order to realise all admissible tripleswith b ≤−2, Halic considers the manifold:

M′ = X ×F2#F1×F2S×F1.

For a given finitely presented group G, let M be the manifold obtained by taking the connectedsum of M′ with the product MG ×F1, where MG is as in Theorem 4.30 and F1 is a surface ofgenus 1. In other words we have

M = M′#F1×F1MG ×F1,

where the sum is performed on the left-hand side along F1 ×F1 ⊂ S×F1 (this is possiblebecause F1 and F2 are disjoint in S) and F1 = T ⊂ MG is also as in (ii) of Theorem 4.30. TheChern numbers of M are the same as those of M′ by Lemma 3.23. Moreover, one can showthat the fundamental group π1(M) is isomorphic to G. To see that, let

U = M′−F1×F1 and V = MG ×F1−F1×F1.

Then U,V is an open covering of M and U ∩V = F1×F1×A. Observe that we may rewriteU as

X ×F2#F1×F2(S−F1)×F1.

Since the complement of F1 in S is simply connected, the argument in [13, p. 379] still goesthrough and shows that U is simply connected. By Seifert-van Kampen, the fundamentalgroup of M is given by

π1(M) = π1(V )/〈π1(U ∩V )〉 = π1(MG −F1)/〈π1(F1)×π1(A)〉. (4.15)


The epimorphismi∗ : π1(MG −F1)→π1(MG),

is surjective because of the codimension of the embedding F1 ⊂ MG. Moreover, the kernel ofi∗ is generated by a meridian of F1 and can be identified with π1(A), so we have

π1(MG −F1)/〈π1(F1 ×A)〉 ∼= π1(MG)/〈π1(F1)〉 ∼= G,

which implies, together with (4.15), that π1(M) ∼= G. By blowing up M we get symplecticrealisations with fundamental group G for all admissible triples with b ≤−2.

Appendix A

Some computations.

A.1 Chern numbers of blow-up in dimension 8.

Recall that this is the situation we are considering: M is a closed, connected, symplectic 8-dimensional manifold, N a symplectic submanifold of M, E denotes the normal bundle of Nin M, P(E) its projectivisation and M the symplectic blow-up of M along N. Notice that if Nis a submanifold of dimension six (i.e., codimension two), then P(E) ∼= N and M ∼= M. Weconsider maps as in the blow-up diagram

P(E)j−−−−→ M

ρE

yy f

Ni−−−−→ M

To carry out the computations, we are going to make use of the Projection Formula, whichstates that for a ∈ H∗(X), b ∈ H∗(Y ) and f : Y→X one has

f !( f ∗(a)∪b) = a∪ f !(b).

This formula is easily proved:

f !( f ∗(a)∪b) = DM f∗D−1N ( f ∗(a)∪b)

= DM f∗(( f ∗(a)∪b)∩ [N])

= DM f∗( f ∗(a)∩ (b∩ [N]))

= DM(a∩ ( f !(b)∩ [M]))

= DM((a∪ f !(b))∩ [M])

= a∪ f !(b).

The Projection Formula immediately implies the following relations:

73

74 APPENDIX A. SOME COMPUTATIONS.

• For a ∈ Hk(M) and b′ ∈ H l(P(E)) we have

f ∗a∪ j!b′ = j!( j∗ f ∗a∪b′) Projection Formula

= j!(ρ∗i∗a∪b′). by commutativity of

the blow-up diagram.

Since i∗a ∈ Hk(N) and the latter is zero for all k > dimN, we get

f ∗a∪ j!b′ = 0 for all a ∈ Hk(M) with k > dimN (A.1)

and similarly, if b′ = ρ∗b for some b ∈ H l(N),

f ∗a∪ j!ρ∗b = 0 for all a ∈ Hk(M),b ∈ H l(N) with k + l > dimN. (A.2)

• Let ξ denote the first Chern class of the dual of the tautological line bundle over P(E).Then

j!ξk ∪ j!ξh = j!(ξk ∪ j∗ j!ξh) Projection Formula

= j!(ξk ∪ (−ξh+1)) Self-Intersection formula

= − j!ξk+h+1.

Moreover, one can prove by induction that

( j!ξ)k = (−1)k+1 j!ξ2k−1.

• If we let η denote the Poincare dual of [P(E)] in M, i.e. η = j!1 = PDM[P(E)], thenwe see that

j∗η = j∗ j!1 = −ξ

by the Self-Intersection formula, and hence, in particular, j!ξk =(−1)kηk+1 and j∗ηk =(−1)kξk.

A.1.1 Blow-up at a point.

If M is blown up at a point, from the blow-up formula (3.14) we get

j∗(c(M)− f ∗c(M)) = −ξ(3ξ3 +2ξ2−2ξ−3),

hencec(M)− f ∗c(M) = j!(3ξ3 +2ξ2−2ξ−3).

The latter we may rewrite as the following system of equations for the individual Chernclasses:

c4(M) = f ∗c4(M)+3 j!ξ3

c3(M) = f ∗c3(M)+2 j!ξ2

c2(M) = f ∗c2(M)−2 j!ξc1(M) = f ∗c1(M)−3η.

A.1. CHERN NUMBERS OF BLOW-UP IN DIMENSION 8. 75

The top-dimensional products of Chern classes are

c4(M) = f ∗c4(M)+3 j!ξ3

c1c3(M) = ( f ∗c1(M)−3η)( f ∗c3(M)+2 j!ξ2)

= f ∗c1c3(M)+2 f ∗c1(M)∪ j!ξ2 −3 f ∗c3(M)∪η−6 j!ξ2 ∪η= f ∗c1c3(M)−6 j!ξ2 ∪η

c22(M) = ( f ∗c2(M)−2 j!ξ)2

= f ∗c22(M)−4 f ∗c2(M)∪ j!ξ+4( j!ξ)2

= f ∗c22(M)−4 j!ξ3

c21c2(M) = ( f ∗c1(M)+−3η)2∪ ( f ∗c1(M)−2 j!ξ)

= f ∗c21c2(M)−2 f ∗c2

1(M)∪ j!ξ+9 f ∗c2(M)∪η2 −18η2∪ j!ξ−6 f ∗c1c2(M)∪η+12 f ∗c1(M)∪η∪ j!ξ

= f ∗c21c2(M)−18 j!ξ3

c41(M) = ( f ∗c1(M)−3η)4

= f ∗c41(M)−12 f ∗c3

1(M)∪η+54 f ∗c21(M)∪η2

−108 f ∗c1 ∪η3 −81η4

= f ∗c41(M)−81η4.

We now proceed to compute the Chern numbers. Recall that in the case of blow-up at apoint, P(E) ∼= CP3. Since ξE = c1(l∗E) is the generator of H∗(CP3) which is compatible withthe orientation, we have

〈 j!ξ3, [M]〉 = −〈η4, [M]〉 = −〈 j∗η3, [P(E)]〉 = 〈ξ3, [P(E)]〉 = 1.

Hence we have:

c4[M] = 〈c4(M), [M]〉= 〈 f ∗c4(M), [M]〉+3〈 j!ξ3

E , [M]〉= c4[M]+3

c1c3[M] = 〈c1c3(M), [M]〉= 〈 f ∗c1c3(M), [M]〉+6〈 j!ξ3, [M]〉= c1c3[M]+6

c22[M] = 〈c2

2(M), [M]〉 (A.3)

= 〈 f ∗c22(M), [M]〉−4〈 j!ξ3, [M]〉

= c22[M]−4

c21c2[M] = 〈c2

1c2(M), [M]〉= 〈 f ∗c2

1c2(M), [M]〉−18〈 j!ξ3, [M]〉= c2

1c2[M]−18


c41[M] = 〈c4

1(M), [M]〉= 〈 f ∗c4

1(M), [M]〉−81〈 j!ξ3, [M]〉= c4

1[M]−81.

A.1.2 Blow-up along a curve.

If M is blown up along a submanifold of dimension 2

j∗(c(M)− f ∗c(M)) = ρ∗c(N)r−1

∑i=0

ci(E(1))(1+ξ)r−i−1(1−ξ)−ρ∗c(E)

= ρ∗c(N)(1+ξ)2(1−ξ)+ c1(E(1))(1+ξ)(1−ξ)

+c2(E(1))(1−ξ)−1−ρ∗c1(E)−ρ∗c2(E)

−ρ∗c3(E)= ρ∗c(N)−2ξ3−ρ∗c1(E)ξ2 +ρ∗c1(E)ξ+2ξ= −ρ∗c(N)2ξ2 +ρ∗c1(E)ξ−ρ∗c1(E)−2ξ.

Remark. In the computations above we have used the fundamental relation

ξ3 +ρ∗c1(E)ξ2 +ρ∗c2(E)ξ+ρ∗c3(E) = 0, (A.4)

which in our situation, namely dimN = 1, reduces to ξ3 +ρ∗c1(E)ξ2 = 0.

Hence

c(M)− f ∗c(M) = j! [ρ∗c(N)(2ξ2 +ρ∗c1(E)ξ−ρ∗c1(E)−2)]

= j! [2ρ∗c1(N)ξ2 +ρ∗c1(E)ξ+2ξ2

−2ρ∗c1(N)−ρ∗c1(E)−2] ,

that is,

c4(M) = f ∗c4(M)+2 j!(ρ∗c1(N)ξ2)

c3(M) = f ∗c3(M)+ j!ρ∗c1(E)ξ+2 j!ξ2

c2(M) = f ∗c2(M)−2 j!ρ∗c1(N)− j!ρ∗c1(E)

c1(M) = f ∗c1(M)−2η.

The top-dimensional products of Chern classes are:

c4(M) = f ∗c4(M)+2 j!ρ∗c1(N)ξ2

c1c3(M) = ( f ∗c1(M)−2η)( f ∗c3(M)+ j!ρ∗c1(E)ξ+2 j!ξ2))

= f ∗c1c3(M)+ f ∗c1(M)∪ j!(ρ∗c1(E)ξ)+2 f ∗c1(M)∪ j!ξ2

−2 f ∗c3(M)∪η−2η∪ j!(ρ∗c1(E)ξ)−4η∪ j!ξ2


c22(M) = ( f ∗c2(M)−2 j!ρ∗c1(N)− j!ρ∗c1(E))2

= f ∗c22(M)+4( j!ρ∗c1(N))2 +( j!ρ∗c1(E))2

−4 f ∗c2(M)∪ j!ρ∗c1(N)−2 f ∗c2(M)∪ j!ρ∗c1(E)

+4 j!ρ∗c1(N)∪ j!ρ∗c1(E)

c21c2(M) = ( f ∗c1(M)−2η)2( f ∗c1(M)−2 j!ρ∗c1(N)− j!ρ∗c1(E))

= f ∗c21c2(M)−2 f ∗c2

1(M)∪ j!ρ∗c1(N)− f ∗c21(M)∪ j!ρ∗c1(E)

−4 f ∗c1c2(M)∪η+8 f ∗c1(M)∪ j!ρ∗c1(N)∪η+4 f ∗c1(M)∪η∪ j!ρ∗c1(E)+4 f ∗c2(M)∪η2

−8η2∪ j!ρ∗c1(N)−4η2 ∪ j!ρ∗c1(E)

c41(M) = ( f ∗c1(M)−2η)4

= f ∗c41(M)−8 f ∗c3

1(M)∪η+24 f ∗c21(M)∪η2

−32 f ∗c1(M)∪η3 +16η4.

Expressions (A.1) and (A.2) immediately imply

f ∗a∪ j!ρ∗b = 0 for all a ∈ H4(M),b ∈ H2(N)

f ∗a∪ j!(ρ∗bξ) = 0 for all a ∈ H2(M),b ∈ H2(N)

f ∗a∪ j!ρ∗b∪η = 0 for all a ∈ H2(M),b ∈ H2(N)

f ∗a∪η = 0 for all a ∈ H6(M)

f ∗a∪η2 = 0 for all a ∈ H4(M).

We now consider the other terms, or rather their evaluation on [M]. We obtain:

〈 j!(ρ∗bξ2), [M]〉 = 〈 j!(ρ∗b j∗η2), [M]〉 = 〈 j!ρ∗bη2, [M]〉= 〈 j∗ j!ρ∗b(−ξ), [P(E)]〉 = 〈ρ∗bξ2, [P(E)]〉= 〈b, [N]〉 for all b ∈ H2(N)

〈 j!(ρ∗bξ)∪η, [M]〉 = 〈 j∗ j!(ρ∗bξ), [P(E)]〉 = 〈−ρ∗bξ2, [P(E)]〉= 〈b, [N]〉 for all b ∈ H2(N)

〈 j!ρ∗b∪η2, [M]〉 = 〈 j∗ j!ρ∗b(−ξ), [P(E)]〉 = 〈ρ∗bξ2, [P(E)]〉= 〈b, [N]〉 for all b ∈ H2(N)

〈 f ∗a∪η3, [M]〉 = 〈 j∗ f ∗a ∪ j∗η2, [P(E)]〉 = 〈ρ∗i∗aξ2, [P(E)]〉= 〈a, [N]〉 for all a ∈ H2(M)

〈 f ∗a∪ j!ξ2, [M]〉 = 〈 f ∗a∪ j!( j∗η2 ∪1), [M]〉 = 〈 f ∗a∪η2∪η), [M]〉= 〈 f ∗a∪η3, [M]〉 = 〈 j∗ f ∗a∪ j∗η2, [P(E)]〉= 〈ρ∗i∗aξ2, [P(E)]〉 = 〈i∗a, [N]〉 for all a ∈ H2(M)

and by the fundamental relation (A.4) we also have

〈 j!ξ2 ∪η, [M]〉 = 〈 j∗ j!ξ2, [P(E)]〉 = 〈ξ3, [P(E)]〉= 〈−ρ∗c1(E)ξ2, [P(E)]〉 = 〈c1(E), [N]〉.


Then the Chern numbers are easily seen to be:

c4[M] = 〈c4(M), [M]〉= c4[M]+2c1[N]

c1c3[M] = 〈c1c3(M), [M]〉= c1c3[M]+2c1[N]

c22[M] = 〈c2

2(M), [M]〉 (A.5)

= c22[M]

c21c2[M] = 〈c2

1c2(M), [M]〉= c2

1c2[M]−8c1[N]−4〈c1(E), [N]〉c4

1[M] = 〈c41(M), [M]〉

= c41[M]−32c1[N]−16〈c1(E), [N]〉.

A.1.3 Blow-up along a four-dimensional submanifold.

Similarly, if M is blown up along a submanifold of dimension 4,

j∗(c(M)− f ∗c(M)) = −ρ∗c(N)(ξ−1)ξ,

hence

c(M)− f ∗c(M) = j!(ρ∗c2(N)ξ+(ρ∗c1(N)ξ−ρ∗c2(N))+(ξ−ρ∗c1(N))−1)

and the Chern classes of the blown up manifold satisfy the following relations

c4(M) = f ∗c4(M)+ j!(ρ∗c2(N)ξ)

c3(M) = f ∗c3(M)+ j!(ρ∗c1(N)ξ)− j!ρ∗c2(N)

c2(M) = f ∗c2(M)+ j!ξE − j!ρ∗c1(N)

c1(M) = f ∗c1(M)−η.

In this case we have the fundamental relation

ξ2 +ρ∗c1(E)ξ+ρ∗c2(E).


For the top-dimensional products of Chern classes we get the expressions:

c4(M) = f ∗c4(M)+ j!(ρ∗c2(N)ξ)

c1c3(M) = ( f ∗c1(M)−η)( f ∗c3(M)+ j!(ρ∗Ec1(N)ξ)− j!ρ∗c2(N))

= f ∗c1c3(M)+ f ∗c1(M)∪ j!(ρ∗c1(N)ξ)− f ∗c1(M)∪ j!ρ∗c2(N)

−η∪ f ∗c3(M)−η∪ j!(ρ∗c1(N)ξ)+η∪ j!ρ∗c2(N)

c22(M) = ( f ∗c2(M)+ j!ξ− j!ρ∗c1(N))2

= f ∗c22(M)+( j!ξ)2 +( j!ρ∗c1(N))2 +2 f ∗c2(M)∪ j!ξ

−2 f ∗c2(M)∪ j!ρ∗c1(N)−2 j!ξ∪ j!ρ∗c1(N)

c21c2(M) = ( f ∗c1(M)−η)2( f ∗c2(M)+ j!ξ− j!ρ∗c1(N))

= ( f ∗c21(M)−2 f ∗c1(M)∪η+η2)( f ∗c2(M)+ j!ξ− j!ρ∗c1(N))

= f ∗c21c2(M)+ f ∗c2

1(M)∪ j!ξ− f ∗c21(M)∪ j!ρ∗c1(N)

−2 f ∗c1c2(M)∪η−2 f ∗c1(M)∪η∪ j!ξ+2 f ∗c1(M)∪η∪ j!ρ∗c1(N)+ f ∗c2(M)∪η2

+ j!ξ∪η2 − j1ρ∗c1(N)∪η2

c41(M) = ( f ∗c1(M)+ j!(−1))4

= f ∗c41(M)−4 f ∗c3

1(M)∪η+6 f ∗c21(M)∪η2

−4 f ∗c1(M)∪η3 +η4.

Again by (A.1) and (A.2) we immediately see that some of the terms which appear abovevanish, namely

f ∗a∪ j!(ρ∗b) = 0 for all a ∈ H2(M) and b ∈ H4(N);

f ∗a∪η = 0 for all a ∈ H6(M).

We evaluate the other terms on the homology class [M]:

〈 f ∗a∪ j!(ρ∗bξ), [M]〉 = 〈 f ∗a∪ j!(ρ∗b(− j∗η)), [M]〉= 〈 f ∗a∪ (− j!ρ∗b)∪η, [M]〉= 〈 j∗ f ∗a∪ (− j∗ j!ρ∗b), [P(E)]〉= 〈ρ∗i∗a∪ρ∗bξ, [P(E)]〉= 〈i∗a∪b, [N]〉

for all a ∈ H2(M),b ∈ H2(N)

〈 f ∗a∪ j!ρ∗b∪η, [M]〉 = 〈 j∗ f ∗a∪ j∗ j!ρ∗b, [P(E)]〉 = 〈ρ∗i∗a∪ (−ρ∗bξ), [P(E)]〉= 〈−ρ∗(i∗a∪b)ξ, [P(E)]〉= −〈i∗a∪b, [N]〉

for all a ∈ H2(M),b ∈ H2(N)


〈 j!(ρ∗bξ)∪η, [M]〉 = 〈 j∗ j!(ρ∗bξ), [P(E)]〉 = 〈−ρ∗bξ2, [P(E)]〉= 〈ρ∗(b∪ c1(E))ξ, [P(E)]〉 = 〈b∪ c1(E), [N]〉

for all b ∈ H2(N)

〈 j!ρ∗b∪η, [M]〉 = 〈−ρ∗bξ, [P(E)]〉 = −〈b, [N]〉for all b ∈ H4(N)

〈( j!ρ∗b)2, [M]〉 = 〈 j!( j∗ j!ρ∗b∪ρ∗b), [M]〉 = 〈 j!(−ρ∗bξ∪ρ∗b), [M]〉= 〈− j!(ρ∗b2 ∪ j∗η), [M]〉 = 〈− j!ρ∗b∪η, [M]〉= 〈i∗a∪b, [N]〉

for all b ∈ H2(N)

〈 f ∗a∪η2, [M]〉 = 〈 j∗ f ∗a∪ j∗η, [P(E)]〉 = 〈ρ∗i∗a∪ (−ξ), [P(E)]〉= −〈i∗a, [N]〉

for all a ∈ H4(M)

〈 j!ρ∗b∪η2, [M]〉 = 〈 j∗ j!ρ∗b∪ j∗η, [P(E)]〉 = 〈−ρ∗bξ∪ (−ξ), [P(E)]〉= 〈−ρ∗(b∪ c1(E))ξ, [P(E)]〉 = −〈b∪ c1(E), [N]〉

for all b ∈ H2(N)

〈 f ∗a∪η3, [M]〉 = 〈 j∗ f ∗a∪ j∗η2, [P(E)]〉 = 〈ρ∗i∗aξ2, [P(E)]〉= 〈−ρ∗(i∗a∪ c1(E))ξ, [P(E)]〉 = −〈i∗a∪ c1(E), [N]〉

for all a ∈ H2(M)

〈η4, [M]〉 = 〈 j∗η3, [P(E)]〉 = 〈−ξ3, [P(E)]〉= 〈ξ(ρ∗c1(E)ξ+ρ∗c2(E)), [P(E)]〉 = 〈−ρ∗c2

1(E)ξ+ρ∗c2(E)ξ, [P(E)]〉= 〈−c2

1(E)+ c2(E), [N]〉.

As to the remaining terms, we simply observe that

( j!ξ)2 = η2

f ∗a∪ j!ξ = − f ∗a∪η2

j!ρ∗b∪ j!ξ = − j!ρ∗b∪η2

f ∗a∪ j!ξ∪η = − f ∗a∪η3

j!ξ∪η2 = −η4.

Using once again the relation 〈 f ∗i!a, [M]〉 = 〈a, [N]〉, we see that the Chern numbers canbe expressed as:

c4[M] = 〈c4(M), [M]〉= c4[M]+ c2[N]

c1c3[M] = 〈c1c3(M), [M]〉= c1c3[M]+ c2

1[N]− c2[N]

A.2. SUBMANIFOLDS FROM DONALDSON’S THEOREM. 81

c22[M] = 〈c2

2(M), [M]〉 (A.6)

= c22[M]− c2

1[N]+2c2[N]−〈c21(E), [N]〉+3〈c2(E), [N]〉

c21c2[M] = 〈c2

1c2(M), [M]〉= c2

1c2[M]− c21[N]− c2[N]−2〈c2(E), [N]〉−2〈c1(N)∪ c1(E), [N]〉

c41[M] = 〈c4

1(M), [M]〉= c4

1[M]−6c21[N]−3〈c2

1(E), [N]〉+ 〈c2(E), [N]〉+−8〈c1(N)∪ c1(E), [N]〉.

A.2 Submanifolds from Donaldson’s theorem.

We refer to the following situation: (S,ωK) is a symplectic sphere bundle with compactsymplectic base (N,β), obtained by projectifying the bundle E ⊕C, with E a complex linebundle over N. The form β is integral and the form ωK is defined as Kρ∗β + η, where ρdenotes the bundle map from S to N and η represents the class ξ ∈ H2(S;Z), which is thefirst Chern class of the dual of the canonical line bundle over S. Then by Theorem 3.10, forlarge enough λ ∈ Z there exist symplectic submanifolds Nλ of S, realising the Poincare dualof the classes λ[ωK ]. Our aim in this section is to compute the characteristic numbers of thesubmanifolds Nλ in the case where the dimension of N equals 4.

Recall from Example 3.6 that the total Chern class of S can be written as:

c(S) = 1+ρ∗(c1(N)+ c1(E))+2ξ+ρ∗(c1(N)∪ c1(E)+ c2(N))+2ρ∗c1(N)ξ+2ρ∗c2(N)ξ.

The Chern classes of S and Nλ are related by the Whitney product formula

c(Nλ)∪ c(νSNλ) = i∗c(S), (A.7)

where νSNλ denotes as usual the normal bundle of Nλ in S, and i the inclusion Nλ→S. Thebundle νSNλ is a complex line bundle and its first Chern class coincides with the Euler class,which in turn is equal to the restriction to Nλ of the Poincare dual of Nλ. Since we knowPD[Nλ] to be λ[ωK ] = λK[ρ∗β]+λξ, we can rewrite equation (A.7) as

c(Nλ)∪ i∗(λK[ρ∗β]+λξ) = i∗c(S) (A.8)

and by substituting the expression for the total Chern class of S we get

c(Nλ)∪ i∗(λK[ρ∗β]+λξ) = (A.9)

i∗[1+ρ∗(c1(N)+ c1(E))+2ξ+ρ∗(c1(N)∪ c1(E)+ c2(N))

+2ρ∗c1(N)ξ+2ρ∗c2(T N)ξ].

By comparing the terms of equal degree on the two sides of the above equation, we get thefollowing two identities:

c1(Nλ)+λKi∗[ρ∗β]+λi∗ξ = i∗ρ∗c1(N)+ i∗ρ∗c1(E)+2i∗ξ


and

c1(Nλ)∪ (λKi∗[ρ∗β]+λi∗ξ)+ i∗c2(Nλ) =

i∗ρ∗c1(N)∪ i∗ρ∗c1(E)+ i∗ρ∗c2(N)+2i∗(ρ∗c1(N)ξ).

Using these identities we can compute the Chern classes of Nλ, namely

c1(Nλ) = i∗ρ∗c1(N)+ i∗ρ∗c1(E)+(2−λ) i∗ξ−λKi∗[ρ∗β]

c2(Nλ) = i∗ρ∗c2(N)+ i∗ρ∗(c1(N)∪ c1(E))+(2−λ)i∗(ρ∗c1(N)ξ)

−λKi∗(ρ∗c1(N)∪ [ρ∗β])− (λ2−λ) i∗(ρ∗c1(E)ξ)

−λKi∗(ρ∗c1(E)∪ [ρ∗β])+ (2λ2K −2λK) i∗([ρ∗β]ξ)

+λ2K2i∗[ρ∗β]2.

In particular, we have

c21(Nλ) = i∗ρ∗c2

1(N)+ i∗ρ∗c21(E)− (λ2−2λ)i∗(ρ∗c1(E)ξ)

+λ2K2i∗[ρ∗β]2 +2i∗ρ∗(c1(N)∪ c1(E))

+2(2−λ) i∗(ρ∗c1(N)ξ)−2λKi∗(ρ∗c1(N)∪ [ρ∗β])

−2λKi∗(ρ∗c1(E)∪ [ρ∗β])−2(2−λ)λKi∗([ρ∗β]ξ).

The top-dimensional Chern classes c21 and c2 can be evaluated on the fundamental ho-

mology class of Nλ. Using Poincare duality we can compute these values. In fact, notice thatc2

1(Nλ) and c2(Nλ) are contained in the image of i∗, i.e., they can be written as i∗x1 and i∗x2,respectively, where each xi is a class of S. But for each x ∈ H4(S;Z), the product 〈i∗x, [Nλ]〉has the form

〈i∗x, [Nλ]〉 = 〈x∪ PD[Nλ], [S]〉 = 〈x∪ (λK[ρ∗β]+λξ), [S]〉.

In order to compute the Chern numbers of Nλ, then, we start by computing the products

x1 ∪ (λK[ρ∗β]+λξ) = λρ∗c21(N)ξ+λ(λ2−2λ+1)ρ∗c2

1(E)ξ+2λ(λ−1)ρ∗(c1(N)∪ c1(E))ξ+4λK(1−λ)ρ∗c1(N)∪ [ρ∗β]ξ+λ2K(4−3λ)ρ∗c1(E)∪ [ρ∗β]ξ+λ2K2(3λ−4)[ρ∗β]2 ξ

x2 ∪ (λK[ρ∗β]+λξ) = λρ∗c2(N)ξ+λ2(λ−1)ρ∗c21(E)ξ

+λ(λ−1)ρ∗(c1(N)∪ c1(E))ξ−2λK(λ−1)ρ∗c1(N)∪ [ρ∗β]ξ+λ2K(2−3λ)ρ∗c1(E)∪ [ρ∗β]ξ+λ2K2(3λ−2)[ρ∗β]2 ξ

Notice that we have used the fundamental relation

ξ2 +ρ∗c1(E)ξ = 0 ∈ H∗(S;Z)

A.2. SUBMANIFOLDS FROM DONALDSON’S THEOREM. 83

in order to reduce the products to this form. From the ring structure of the cohomology of Swe also get

〈ρ∗yξ, [S]〉 = 〈y, [N]〉 (A.10)

for all y ∈ H4(N;Z). We can thus immediately write down the Chern numbers of Nλ, whichare

c21[Nλ] = λc2

1[N]+λ(λ2−2λ+1)〈c21(E), [N]〉

+2λ(λ−1)〈(c1(N)∪ c1(E)), [N]〉+4λK(1−λ)〈c1(N)∪ [ρ∗β], [N]〉+λ2K(4−3λ)〈c1(E)∪ [ρ∗β], [N]〉+λ2K2(3λ−4)〈[ρ∗β]2, [N]〉

c2[Nλ] = λc2[N]+λ2(λ−1)〈c21(E), [N]〉

+λ(λ−1)〈c1(N)∪ c1(E), [N]〉−2λK(λ−1)〈c1(N)∪ [ρ∗β], [N]〉+λ2K(2−3λ)〈c1(E)∪ [ρ∗β], [N]〉+λ2K2(3λ−2)〈[ρ∗β]2, [N]〉.

We are also interested in other invariants of Nλ, namely 〈c21(νSNλ), [Nλ]〉 and 〈c1(Nλ)c1(νSNλ), [Nλ]〉.

We use the same strategy as for c21 and c2. Recall that c1(νSNλ) = i∗(λξ+λK[ρ∗β]): then

〈c21(νSNλ), [Nλ]〉 = 〈i∗(λξ+λK[ρ∗β])2, [Nλ]〉 = 〈(λξ+λK[ρ∗β])3, [S]〉

= 〈λ3ξ3 +3λ3K[ρ∗β]ξ2 +3λ3K2[ρ∗β]2ξ, [S]〉= 〈λ3ρ∗c2

1(E)ξ+3λ3Kc1(E)∪ [ρ∗β]ξ+3λ3K2[ρ∗β]2ξ, [S]〉

= λ3〈c21(E), [N]〉+3λ3K〈c1(E)∪ [β], [N]〉

+3λ3K2〈[β]2, [S]〉.

Similarly we compute the last invariant:

〈c1(Nλ)∪ c1(νSNλ), [Nλ]〉 = 〈i∗(ρ∗c1(N)+ρ∗c1(E)+(2−λ)ξ−λK[ρ∗β])

∪ i∗(λξ+λK[ρ∗β]), [Nλ]〉= 〈(ρ∗c1(N)+ρ∗c1(E)+(2−λ)ξ−λK[ρ∗β])

∪(λξ+λK[ρ∗β])2, [S]〉= 〈λ2c1(N)ξ2 +λ2c1(E)ξ2 +λ2(2−λ)ξ3

−λ3K[ρ∗β]ξ2 +2λ2Kc1(N)[ρ∗β]ξ+2λ2Kc1(E)[ρ∗β]ξ+2(2−λ)λ2K[ρ∗β]ξ2

+λ2K2(2−λ)[ρ∗β]2ξ−2λ3K2[ρ∗β]2ξ, [S]〉


= 〈−λ2c1(N)∪ c1(E)ξ−λ2c21(E)ξ

+λ2(2−λ)c21(E)ξ−λ3Kc1(E)∪ [ρ∗β]ξ

+2λ2Kc1(N)[ρ∗β]ξ+2λ2Kc1(E)[ρ∗β]ξ+2(2−λ)λ2K c1(E)∪ [ρ∗β]ξ+λ2K2(2−λ)[ρ∗β]2ξ−2λ3K2[ρ∗β]2ξ, [S]〉

= −λ2〈c1(N)∪ c1(E), [N]〉+λ2(1−λ)〈c2

1(E), [N]〉+λ2K(3λ−2)〈c1(E)∪ [β], [N]〉+2λ2K〈c1(N)[β], [N]〉+λ2K2(2−3λ)〈[β]2, [N]〉.

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Samenvatting.

Over de geografie van symplectische varieteiten.

Het onderwerp van dit proefschrift betreft constructies en invarianten van symplectischevarieteiten.

Een symplectische varieteit is een gladde varieteit met een symplectische structuur. Datwil zeggen een gesloten, niet-gedegenereerde 2-vorm. Een belangrijk voorbeeld van eendergelijke varieteit komt uit de Hamiltoniaanse mechanica, namelijk de totaalruimte vande coraakbundel van een gladde varieteit. Dit kan men ook beschouwen als de faseruimtevan een dynamisch systeem over de gegeven gladde varieteit. Een diffeomorfisme tussentwee symplectische varieteiten dat de symplectische structuur behoudt, heet symplectomor-fisme. Deze symplectomorfismen behouden tevens de klasse van Hamiltoniaanse differen-tiaalvergelijkingen.

Twee van de meest opvallende vragen in verband met symplectische meetkunde betref-fen het bestaan van symplectische structuren en de definitie van symplectische invarianten.Aangezien volgens de stelling van Darboux alle symplectische structuren lokaal isomorf zijn,moeten deze invarianten globaal van aard zijn.

Dit proefschrift gaat over enkele numerieke invarianten van symplectische varieteiten.Gegeven een gesloten, 2n-dimensionale varieteit met een symplectische vorm ω, dan bestaater over haar raakbundel een complexe structuur. Ten opzichte van deze structuur is het mo-gelijk een systeem van π(n) gehele getallen te bepalen, waarbij π(n) de cardinaliteit vande verzameling van alle partities van n is. Deze getallen zijn alleen van de symplecti-sche structuur afhankelijk en worden Chern getallen genoemd, naar de Chinees-Amerikaanswiskundige Shiing-Shen Chern, geboren in 1911.

Het specifieke probleem waarmee wij ons in dit proefschrift bezighouden is dat van welkesystemen van gehele getallen mogen optreden als systeem van Chern getallen van een ge-sloten, samenhangende, symplectische varieteit. Dit probleem is bekend in de literatuur onderde naam “symplectische geografie”.

De Chern getallen van een willekeurige symplectische varieteit voldoen aan bepaaldevoorwaarden, die uit de algebraische meetkunde zijn ontstaan, namelijk een aantal congruen-tie-relaties die de stelling van Riemann-Roch impliceert. Het hoofdresultaat van dit proef-schrift laat zien, dat in dimensie 8 deze voorwaarden ook voldoende zijn. Dat wil zeggen,gegeven een vijftal van gehele getallen (a1, . . . ,a5) die aan de congruentie-relaties voldoen,dan bestaat er een gesloten, samenhangende, symplectische 8-dimensionale varieteit M, zo

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dat (a1, . . . ,a5) met het vijftal van Chern getallen van M overeenstemt.De inhoud van het proefschrift is als volgt. Hoofdstuk 1 bevat de belangrijkste defini-

ties en feiten uit de literatuur. Hoofdstuk 2 gaat over verschillende manieren om symplec-tische varieteiten te construeren. In het derde hoofdstuk beschouwen wij de symplectischeinvarianten en we beschrijven, hoe zij efficient te berekenen zijn voor de varieteiten die wijin hoofdstuk 2 hebben geconstrueerd. De details van de berekeningen zijn in de appendixuitgevoerd. In hoofdstuk 4 construeren wij voorbeelden van 8-dimensionale symplectischevarieteiten en we realiseren zo alle vijftallen die door de congruentie-relaties niet zijn uitges-loten. Daarmee is ons hoofdresultaat bewezen.

Documents

On the geography of symplectic manifolds · Then we recall Darboux’s theorem, which states that any two symplectic forms are locally isomorphic, so that symplectic invariants must