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Examples of Symplectic Manifolds

Bruijnen L. - Lorenzin A. - de Maat P. - Sitbon P. - Zacharopoulos G.

February 4, 2019

Abstract

The aim of this paper is to present some historical constructions of closed

symplectic manifolds that are not Kähler following the lead of [6]. After presenting symplectic manifolds, Kähler manifolds and related notions, we follow

the first example of symplectic non Kähler manifold given by Thurston (1976)

[7]. The construction of this example, later investigated in the context of symplectic fibrations, can be generalized to provide other examples. In the end, we

prove a deep theorem first shown by Gompf (1995) [1], that provides a large

number of counterexamples as the odd Betti numbers of a Kähler manifold

have to be even.

1

Introduction

Symplectic Geometry has its roots in the Hamiltonian formulation of Classical Mechanics and hence it plays an important role in Physics. One of the most important

examples of symplectic manifold is the cotangent space of a smooth manifold M . If

the manifold M represents the set of possible positions in a dynamical system, then

the cotangent bundle, T ∗ M , can be thought of as the set of possible positions and

momenta.

Symplectic manifolds are smooth manifolds together with a closed nondegenerate 2-form ω, which implies that they have even dimension. In the case of closed

symplectic manifold (compact without boundary), the cohomology class of ω and

its powers are nontrivial. This does not happen in general, the cotangent space

being an example where the symplectic form is exact. A rather important class of

symplectic manifolds is the one of Kähler manifolds, which admit a complex structure satisfying some compatibility requirement. In fact, Kähler manifolds are the

standard manifolds appearing in Complex Geometry.

As a matter of fact, initially people were not interested in closed symplectic

manifolds, as they were not the ones appearing in Classical Mechanics. Over time,

interest in closed symplectic manifolds increased, hence there have been many publications about them and their relation to Kähler manifolds. H. Guggenheimer [2, 3]

suggested that closed symplectic manifolds have even odd Betti numbers, a property

that was already proven for closed Kähler manifolds. If this would be true, there

would be restrictions on the fundamental group of a closed symplectic manifold.

Guggenheimer’s claim was later disproved by Thurston (1976) [7], together with the

first example of a closed symplectic non Kähler manifold (Example 5.1) with first

cohomology group Z3 . Thurston’s example is a particular case of a surface fibration

1

over a symplectic manifold. His argument can be extended to the more general

setting of symplectic fibrations:

Theorem 6.3. Let π : M → B be a compact symplectic fibration with symplectic

fibre (F, σ) and connected symplectic base (B, β). Denote by σb ∈ Ω2 (Fb ) the canonical symplectic form on the fibre Fb and suppose that there is a cohomology class

a ∈ H 2 (M ) such that

i∗b a = [σb ]

for all b ∈ B. Then, for every sufficiently large real number K > 0, there exists a

symplectic form ωK ∈ Ω2 (M ) which is compatible with the fibration π and represents

the class a + K[π ∗ β].

In 1995, Gompf provided a groundbreaking result that will also be proved in this

paper:

Theorem 8.1. Let G be a finitely presented group. Then there exists a compact

symplectic 4-manifold (i.e. a manifold of real dimension 4) with fundamental group

G.

Using the fact that odd Betti numbers of a Kähler manifold are even, this provides a large number of symplectic manifolds that are not Kähler.

We finish with a brief discussion about blow up techniques and present the construction of the first simply connected symplectic non Kähler manifold by MacDuff

[5] (1984), obtained blowing up the Kodaira-Thurston manifold (Example 5.1).

2

Symplectic Manifolds

In this document, every manifold is assumed to be connected, smooth, without

boundary. We recall that a closed manifold is a compact manifold without boundary,

so under our assumptions on M , closed and compact will be synonyms.

Definition 2.1. A symplectic structure on a smooth manifold M is a 2-form ω ∈

Ω2 (M ) such that it is:

• nondegenerate: for any q ∈ M , whenever ω(v, w) = 0 for any w ∈ Tq M , then

v = 0.

• closed : d ω = 0.

The pair (M, ω) is called symplectic manifold.

Lemma 2.2. If a smooth manifold M admits a nondegenerate 2-form ω ∈ Ω2 (M ),

it is necessarily even dimensional and oriented. In particular, the orientation of M

is given by the volume form

ωn = ω ∧ · · · ∧ ω

where n is half the dimension of M .

The even dimension comes from the fact that any skew symmetric matrix must

have a kernel in odd dimension. A direct corollary is:

2

Corollary 2.3. A symplectic manifold (M, ω) is necessarily even dimensional and

orientable.

Example 2.4. Given the coordinates (x1 , . . . , xn , y1 , . . . , yn ) on R2n , the following

form defines a symplectic structure on R2n

ω0 =

n

X

d xj ∧ d y j

j=1

We can also view ω0 as a nondegenerate skew-symmetric bilinear form ω0 : R2n ×

R2n → R by identifiying Tx R2n and R2n for all x ∈ R2n . Explicitly, considered

ζ = (ξ, η), ζ 0 = (ξ 0 , η 0 ) ∈ R2n , the form ω0 reads

0

ω0 (ζ, ζ ) =

n

X

(ξk ηk0 − ηk ξk0 ) = −ζ t J0 ζ 0

k=1

where

J0 =

0 −1

1 0

Example 2.5. On S 2 we can define the following nondegenerate closed 2-form:

ωx (u, v) = hx, u × vi

This form is bilinear and skew-symmetric; it is also closed because it is a 2-form on

S 2 , manifold of dimension 2.

Figure 1: Symplectic structure on S 2

Definition 2.6. A symplectomorphism between two symplectic manifolds (M, ω)

and (N, σ) is a diffeomorphism ϕ : M → N which preserves the symplectic form,

i.e. ϕ∗ σ = ω.

Example 2.7. Let us now look at the n-dimensional torus Tn , that is an important

example of symplectic manifold in our context. Given usual coordinates (x1 , . . . , x2n )

on R2n , the symplectic form

ω=

n

X

d x2k−1 ∧ d x2k

k=1

is invariant under translations in R2n , which means that translations are symplectomorphisms. Hence if p : R2n → T2n is the quotient map induced by the actions of

translation by integers, then ω defines a symplectic form σ such that,

p∗ σ = ω

3

Example 2.8. Any orientable smooth manifold M of real dimension 2 is symplectic.

Indeed if ω is a nowhere vanishing 2-form, in dimension two it is automatically

nondegenerate. It is also closed because M has dimension 2, so (M, ω) is a symplectic

manifold. In particular, Riemann surfaces (which are complex manifolds of real

dimension 2) are symplectic.

Remark 2.9. Let us notice some interesting properties on symplectic manifolds:

• From nondegeneracy, we obtain a canonical isomorphism

T M → T ∗ M : X 7→ ι(X)ω = ω(X, ·)

• The closedness requirement entails Rthat [ω] ∈ H 2 (M ; R). In the case of M

closed, since ω n is a volume form, ω n 6= 0, which is the same to say that

[ω n ] ∈ H 2n (M ; R) is nontrivial, in particular [ω k ] ∈ H 2k (M ; R) is nontrivial

for all k ≤ n. We conclude that every closed symplectic manifold M has even

cohomology group H 2k (M ; R) 6= 0 for any k ≤ n. In particular, this entails

that S 2n is not a symplectic manifold for any n > 1.

3

Complex structures

In order to define Kähler manifolds, we need to focus on the relation between symplectic forms and almost complex structures. Theorem 3.6 is rather important in

this framework: it states that every symplectic manifold is almost complex. After¯ Later, these

wards, we will introduce complex manifolds and the operators ∂, ∂.

objects will become crucial to define a symplectic (“associated Kähler”) form on the

projective space.

Definition 3.1. An almost complex structure on a smooth manifold M is an automorphism J of the tangent bundle T M such that J 2 = −Id. We call an almost

complex manifold a pair (M, J) where M is a smooth manifold and J an almost

complex structure on M . We denote the space of almost complex structures on a

smooth manifold M by:

J (M ) := {J ∈ C ∞ (M, End(T M )) | J 2 = −Id}

Lemma 3.2. Every almost complex manifold is even dimensional and oriented.

Proof. Let us consider a Riemannian metric h and define ∀q ∈ M , ∀X, Y ∈ Tq M ,

g(X, Y ) = h(X, Y ) + h(JX, JY )

The Riemannian metric g verifies g(X, Y ) = g(JX, JY ). Hence we can define a

two form ω ∈ Ω2 (M ) by ω(X, Y ) = g(JX, Y ). The form ω is skew-symmetric and

nondegenerate, hence from Lemma 2.2 we conclude that M is oriented and even

dimensional.

Definition 3.3. Consider g a Riemannian metric on M , J ∈ J (M ) and ω ∈ Ω2 (M )

a nondegenerate form. The metric g is J-compatible if g(X, Y ) = g(JX, JY ) for any

q ∈ M and X, Y ∈ Tq M .

4

The almost complex structure J is ω-compatible if

ω(JX, JY ) = ω(X, Y )

for any q ∈ M and X, Y ∈ Tq M

and if

ω(X, JX) > 0

for any nonzero vector X.

Example 3.4. The standard almost complex structure J0 on R2n defined as

∂

∂

∂

∂

=

,

J0

=−

J0

∂xj

∂yj

∂yj

∂xj

is ω0 -compatible.

Definition 3.5. J ∈ J (M ) is an integrable almost complex structure (or briefly,

complex structure) if M can be covered by charts φ : U → φ(U ) ⊂ R2n such that

d φ(q) ◦ Jq = J0 ◦ d φ(q) for any q ∈ U .

Next, let us study the relations between the following structures:

• a Riemannian metric g : T M × T M → R;

• a nondegenerate 2-form ω ∈ Ω2 (M );

• an almost complex structure J : T M → T M .

We have already seen that a compatible metric g and an almost complex structure

J can be used to define a nondegenerate 2-form ω, and that we can create a Jcompatible metric g using any metric h (see proof of Lemma 3.2). We have also

seen what it means for J and ω to be compatible, and it is clear that this definition

entails g(X, Y ) := ω(X, JY ) to be a Riemann metric. In all these cases, we have used

two structures to construct the third. Similarly, we can create an almost complex

structure using a pair (ω, g).

Theorem 3.6. Let (M, ω) be a symplectic manifold. Then there exists an ωcompatible almost complex structure J on M .

Proof. Let g be a Riemannian metric. Since v 7→ g(v, ·) and v 7→ ω(v, ·) are isomorphisms Tq M → Tq∗ M for q ∈ M (by nondegeneracy), we can define an isomorphism

A : T M → T M such that ω(·, ·) = g(A·, ·). Note that g(Av, w) = ω(v, w) =

−ω(w, v) = g(−Aw, v) = g(v, −Aw) so the conjugate of A is −A. Apply polar decomposition on A to get A = U Q for some U, Q : T M → T M where U is orthogonal

and Q is symmetric positive definite. Then Q is well-defined (in a smooth manner)

and satisfies Q2 = −A2 . Since −A commutes with A, Q also commutes with A, so

2

defining Jg,ω = Q−1 A = Q−1 U Q : T M → T M gives Jg,ω

= (−A2 )−1 A2 = − Id.

By construction, Jg,ω is an almost complex structure. Moreover, if ω was constructed using g and some almost complex structure J, then Jg,ω = J, and analogously with the roles of g and ω inverted. In this case, g and ω are said to be

compatible.

Since we will be studying symplectic manifolds, we will always have an almost

complex structure compatible with the symplectic form. As we will see more precisely in the next section, the property that needs to fail on a symplectic non Kähler

manifold is in fact the integrability of the almost complex structure.

Let us recall some definitions regarding complex manifolds.

5

Definition 3.7. A holomorphic atlas on a differentiable manifold of dimension 2n

is an atlas {(Ui , φi )} such that φi (Ui ) ⊆ Cn and the transition functions φij :=

φi ◦ φ−1

: φj (Ui ∩ Uj ) → φi (Ui ∩ Uj ) are holomorphic. The pair (Ui , φi ) is called

j

0

0

holomorphic chart. Two holomorphic atlases {(Ui , φi )}, {(Uj , φj )} are equivalent if

0

0

0

0

all maps φi ◦ φj−1 : φj (Ui ∩ Uj ) → φi (Ui ∩ Uj ) are holomorphic.

Definition 3.8. A complex manifold X of dimension n is a real differentiable manifold of dimension 2n endowed with an equivalent class of holomorphic atlases.

Example 3.9 (Complex projective space CP n ). CP n is defined to be the space

of all complex lines in Cn+1 that meet the origin. We can use affine coordinates

to provide holomorphic transition maps. If z = (z 0 , . . . , z n ) ∈ Cn+1 − {0}, then [z]

corresponds to the line generated by (z 0 , . . . , z n ), i.e. [z] = {(λz 0 , . . . , λz n ) : λ ∈ C}.

Sometimes we will also write (z 0 : · · · : z n ) for [z].

Let Ui = {[z] ∈ CP n |zi 6= 0}. The affine charts φi : Ui → Cn are defined as

follows:

1

φi ([z]) = i (z 0 , . . . , zˆi , . . . , z n )

z

Each chart is well defined and bijective since each [z] intersects with the hyperplane

{z i = 1} exactly in one point. Moreover the transition maps for j > i act as follows:

!

i−1

i

n−1

0

bj

z

1

z

z

z

z

0

n−1

,..., j , j, j,..., j,..., j

φj ◦ φ−1

)=

i (z , . . . , z

zj

z z z

z

z

Hence they are holomorphic.

Definition 3.10. Let X be a complex manifold of complex dimension n and Y ⊂ X

be a differentiable submanifold of real dimension 2k. Then Y is a complex submanifold if there exists a holomorphic atlas {(Ui , φi )} of X such that φi (Ui ∩Y ) ' φi (Ui )∩

Ck . Here, Ck is embedded into Cn via the inclusion (z1 , ..., zk ) 7→ (z1 , ..., zk , 0, ..., 0).

The codimension of Y in X is by definition dim(X) − dim(Y ) = n − k.

Our next goal is to define the complex analogue of the differential of real manifolds as discussed in [4]. Let U ⊂ Cn be an open subset. We can consider U as a

2n-dimensional real manifold with the standard coordinates (x1 , . . . , xn , y1 , . . . , yn ).

For each p ∈ U we have the real tangent space Tp U which is of real dimension 2n.

A canonical basis of Tp U is given by the tangent vectors

∂

∂

∂

∂

|p , . . . ,

|p ,

|p , . . . ,

|p

∂x1

∂xn ∂y1

∂yn

where z1 = x1 + ıy1 , . . . , zn = xn + ıyn are the standard coordinates of Cn . Each

tangent space Tp U admits a natural almost complex structure defined by

J : Tp U → Tp U,

∂

∂

|p 7→

|p ,

∂xi

∂yi

∂

∂

|p 7→ −

|p .

∂yi

∂xi

So we have the following:

Proposition 3.11. The tangent bundle TC U := T U ⊗ C decomposes as a direct sum

of complex vector bundles

TC U = T 1,0 U ⊕ T 0,1 U

The vector bundles T 1,0 U and T 0,1 U are trivialized by the sections

and ∂∂z¯j = 12 ( ∂x∂ j + ı ∂y∂ j ).

6

∂

∂zj

= 12 ( ∂x∂ j −ı ∂y∂ j )

The cotangent bundle TC∗ U := T ∗ U ⊗ C admits an analogous decomposition

= T ∗ U 1,0 ⊕ T ∗ U 0,1 where T ∗ U 1,0 and T ∗ U 0,1 are trivialized by the dual basis

d z := d xj + ı d y j and d z¯j := d xj − ı d y j respectively.

TC∗ U

j

Proposition 3.12. Let f : U → V be a holomorphic map between open subsets U ⊂

Cm and V ⊂ Cn . Then the C-linear extension of the differential d f : Tx U → Tf (x) V

0,1

0,1

respects the above decomposition, i.e. d f (Tx1,0 U ) ⊂ Tf1,0

(x) V and d f (Tx U ) ⊂ Tf (x) V .

Similarly, we can decompose the bundles of k-forms.

Definition 3.13. Let U ⊂ Cn be an open subset. Over U we define the complex

vector bundles

Λp,q U := Λp ((T ∗ U )1,0 ) ⊗ Λq ((T ∗ U )0,1 )

Also we denote by ΩkC (U ) and Ωp,q (U ) the space of sections of ΛkC U and Λp,q U

respectively.

L

p,q

Therefore we have the natural decomposition ΩkC (U ) =

p+q=k Ω (U ) of the

space of differential forms.

As a result, we can define the operators

n

1X ∂

∂

∂=

−ı

(dxj + ıdy j )

2 j=1 ∂xj

∂yj

and

n

1X

∂¯ =

2 j=1

∂

∂

+ı

∂xj

∂yj

(dxj − ıdy j )

where ∂ : Ωp,q (U ) → Ωp+1,q (U ) and ∂¯ : Ωp,q (U ) → Ωp,q+1 (U ).

4

Kähler Manifolds

We can adopt two different points of view to define Kähler manifolds: a symplectic

or a complex approach. Recall that if ω is a nondegenerate 2-form on a manifold M

then there exists an almost complex structure J such that g(X, Y ) := ω(X, JY ) is

a Riemannian metric compatible with the almost complex structure, i.e. it satisfies

g(JX, JY ) = g(X, Y ) (cf. Theorem 3.6). Moreover, J is compatible with the

symplectic form.

Definition 4.1 (Symplectic approach). A Kähler manifold is a symplectic manifold (M, ω) equipped with an integrable almost complex structure J which is ωcompatible.

Definition 4.2 (Complex approach). Let M be a complex manifold with complex

structure J and compatible Riemannian metric g. If the alternating 2-form

ω(X, Y ) := g(JX, Y )

is closed, M (together with g) is a Kähler manifold. Further, ω is called the associated Kähler form and g is a Kähler metric.

Example 4.3. (R2n , J0 , ω0 ) is a Kähler manifold.

7

To check integrability of an almost complex structure we have a powerful theorem

using the Nijenhuis tensor field of J.

Theorem 4.4. Let M be a smooth manifold with almost complex structure J. Define

the Nijenhuis tensor field of J as:

NJ (X, Y ) := [X, Y ] + J[JX, Y ] − J[X, JY ] − [JX, JY ]

The integrability of J is equivalent to the vanishing of NJ .

Corollary 4.5. Let (M, J) be an almost complex manifold of real dimension 2.

Then J is integrable and hence it defines a complex structure on M .

Proof. For any p ∈ M , we can find a nowhere vanishing smooth vector field X in

a neighborhood of p. {X, JX} is a local frame and direct computations show that

NJ (X, JX) = 0. Also, NJ is skew-symmetric, so this is sufficient to conclude that

NJ vanishes on this local frame. As p was chosen arbitrarily NJ = 0, hence J is

integrable.

Since any symplectic manifold has a compatible almost complex structure (Theorem 3.6), the following result holds:

Corollary 4.6. If (M, ω) is a symplectic manifold of dimension 2 then M is a

Kähler manifold.

This is an important result in our search of symplectic manifolds that are not

Kähler: it says that we have to look into at least 4-dimensional manifolds. Also we

already saw in Example 2.8 that Riemann surfaces are symplectic, so we have the

following corollary:

Corollary 4.7. Riemann surfaces are Kähler manifolds.

Example 4.8. Products of Kähler manifolds, endowed with the product complex

structure and the product metric, are Kähler manifolds.

Example 4.9. The complex projective space CP n carries a canonical Kähler metric,

the Fubini-Study metric. Let {Ui : i = 0, . . . , n} be the standard affine open covering

of CP n defined as in Example 3.9. On each Ui , we define

!

n

X

zk 2

ı ¯

.

ωi (z0 : · · · : zn ) :=

∂ ∂ log

zi

2π

k=0

The image of this form under the map φi is

n

X

ı ¯

∂ ∂ log(

|wl |2 + 1)

2π

l=1

z −1

where wj = φi ( jzi ). Those ωi glue to a global form ωF S . This comes from the fact

that ωi |Ui ∩Uj = ωj |Ui ∩Uj . Indeed

!

!

2 n 2 !

2 !

n

n

X

X

zk 2

zk 2

zj X zk

zj

= log

= log + log

.

log

zi

zi

zj

zi

zj

k=0

k=0

k=0

8

2

z

¯

Therefore, it is enough to show that ∂ ∂ log( zji ) = 0 on Ui ∩ Uj , but since

j-th coordinate function on Ui this follows from the fact that

1 ¯

zd¯

z

d¯

z

2

¯

∂ ∂ log|z| = ∂

∂(z z¯) = ∂

=∂

= 0.

z z¯

z z¯

z¯

zj

zi

is the

¯ = −∂ ∂¯ so this

Also, the Fubini Study metric ωF S is a real form. Indeed, ∂ ∂¯ = ∂∂

implies ωi = ω

¯ i . In order to prove that ωF S is a Kähler P

form we have to prove that

is closed and positive defined. Firstly, ∂ωi = 2πı ∂ 2 ∂¯ log( nk=0 | zzki |2 ) = 0, so for each

i, ωi is closed and therefore ωF S is a closed form. To prove that is positive defined,

notice that for each φi (Ui ) we have

! P

P

P

n

X

dwi ∧ dw¯i ( w¯i dwi ) ∧ ( wi dw¯i )

2

¯

P

P

∂∂ 1 +

|wi | =

−

1 + |wi |2

(1 + |wi |2 )2

i=1

X

1

P

hij dwi ∧ dw¯j ,

=

(1 + |wi |2 )2

P

with hij = (1 + |wi |2 )δij − w¯i wj . The matrix (hij ) is positive defined since

ut (hij )¯

u = (u, u) + (w, w)(u, u) − ut ww

¯ t u¯ = (u, u) + (w, w)(u, u) − (u, w)(u, w)

= (u, u) + (w, w)(u, u) − |(w, u)|2 > 0

and the inequality holds from the Cauchy-Schwarz inequality.

Proposition 4.10. Every complex submanifold N of a Kähler manifold M is Kählerian with respect to the induced Riemannian metric.

Proof. If N is a complex submanifold of the Kähler manifold M , the inclusion

i : N → M is a holomorphism, hence the complex structure on N is just the

restriction of the complex structure on M :

JM ◦ i∗ = i∗ ◦ JN .

The induced Riemannian metric on N is still Hermitian and the Kähler form on N

is obtained by restricting the Kähler form on M , hence it is still closed:

d(i∗ ω) = i∗ (dω) = 0.

So N is a Kähler manifold.

Corollary 4.11. Any complex submanifold of CP n (and of R2n ) is a Kähler manifold.

These results show how difficult it can be to produce an example of a symplectic

manifold which is not Kähler. In order to find such a manifold, the easiest way is

to consider topological invariants. First of all, it is well-known that this cannot be

done with local invariants, because symplectic manifolds are locally diffeomorphic.

Theorem 4.12 (Darboux). Let (M, ω) be a symplectic manifold of dimension 2n.

Then ω is locally diffeomorphic to the standard ω0 on R2n .

9

Therefore it is necessary to focus on global invariants.

Definition 4.13. For any k ∈ N,

bk (M ) := dimC H k (M ; C) = dimR H k (M ; R) = rank H k (M ; Z)

is called Betti number of the manifold M .

The Hodge Decomposition, whose statement and proof involves studies on several

topics that here will not be discussed, gives us an incredibly powerful tool:

Proposition 4.14. Let M be a compact Kähler manifold. Then, if k is odd, the

Betti number bk (M ) is even.

In fact, we will use this result in all the examples of symplectic non Kähler

manifolds we will consider.

5

Kodaira – Thurston example

Initially it was unclear if every closed symplectic manifold had a Kähler structure.

The first example was known to Kodaira, and later rediscovered by Thurston.

Example 5.1 (Kodaira – Thurston). Let us consider the set Γ = Z2 × Z2 equipped

with the non abelian internal composition law:

1 i1

(i, j) ◦ (k, l) = (i + k, Ai l + j),

Ai :=

0 1

where i = (i1 , i2 ). This operation gives an isomorphism between (Γ, ◦) and a subgroup of the matrices 5 × 5 with integer coefficients, via the following map:

1 i1 j1 0 0

0 1 j2 0 0

((i1 , i2 ), (j1 , j2 )) 7→

0 0 1 0 0

0 0 0 1 i 2

0 0 0 0 1

Consider the symplectic manifold (R4 , ω), where ω = dx1 ∧ dx2 + dy1 ∧ dy2 , and the

action of Γ onto R4 defined as:

ϕ:

Γ × R4

−→

R4

((i, j), (x, y)) 7→ (x + i, Ai y + j)

This action preserves the symplectic form ω in the sense that each map

ϕ(i,j) :

R4 −→

R4

(x, y) 7→ (x + i, Ai y + j)

is a symplectomorphism. Indeed it is clear that each ϕ(i,j) is a diffeomorphism and

the pullback of ω via ϕ(i,j) is ω. We have that

ω(h, k) = (dx1 ∧ dx2 )(h, k) + (dy1 ∧ dy2 )(h, k)

= (h11 k12 − h12 k11 ) + (h21 k22 − h22 k21 )

10

and since d(x,y) ϕ(i,j) (h1 , h2 ) = (h1 , Ai h2 ),

ϕ∗(i,j) (ω)(h, k) = (h11 k12 − h12 k11 ) + ((h21 + jh22 )k22 − h22 (k21 + jk22 ))

= (h11 k12 − h12 k11 ) + (h21 k22 − h22 k21 )

= ω(h, k).

Hence M = R4 /Γ is a compact symplectic manifold. Its fundamental group is

π1 (M ) = Γ, from which H1 (M, Z) = (π1 (M ))ab = Γ/[Γ, Γ]. Notice [Γ, Γ] ∼

= 0⊕0⊕

Z⊕0 ∼

j)

=

(−i,

−A−i j),

= Z. Indeed, since for any (i, j) ∈ Γ, (i, j)−1 = (−i, −A−1

i

we have:

(i, j) ◦ (h, k) ◦ (i, j)−1 ◦ (h, k)−1 = (0, (i1 l1 − k1 j1 , 0)).

We can conclude that H1 (M, Z) ∼

= Z3 has rank 3, i.e. b1 (M ) = 3. From Proposition

4.14, M does not admit a Kähler structure. Another formulation of this example

can be given by using symplectic fibration, that we are going to define in the next

section.

6

Symplectic Fibrations

Symplectic fibrations are a particular case of fiber bundles. Under certain conditions, it is possible to construct an associated symplectic form, so that a symplectic

fibration will become a symplectic manifold (cf. Theorem 6.3). Such a result, due to

Thurston, allows us to consider manifolds constructed similarly to Kodaira-Thurston

example.

Definition 6.1. Let F be a smooth manifold. A locally trivial fibration with fibre

F is a smooth map π : M → B between smooth manifolds equipped with an open

cover {Uα } of B and a collection of diffeomorphisms φα : π −1 (Uα ) → Uα × F such

that the following diagram commutes for every α:

φα

π −1 (Uα )

Uα × F

pr1

π

Uα

The maps φα are called local trivializations. We denote by Fb := π −1 (b) the fibre

over b ∈ B and by φα (b) : Fb → F the restriction of φα to Fb followed by the

projection onto F . The maps φβα : Uβ ∩ Uα → Diff(F ) defined by

φβα (b) = φβ (b) ◦ φα (b)−1

are called the transition functions. For each b ∈ B the inclusion of the fibre is

denoted by ib : Fb → M .

11

fibre

base

fibre bundle

It can happen that the transition functions takes their value into a specific group,

as for instance for vector bundle where they take their value in GL(k, R) or GL(k, C).

In these cases we talk about the structure group G of the fibration.

In the context of symplectic fibration one restricts to G = Symp(F, σ), where

(F, σ) is a compact symplectic manifold. In this case π : M → B is called a

symplectic fibration and the fibers Fb inherit a symplectic form σb = φα (b)∗ σ.

Definition 6.2. A form ω ∈ Ω2 (M ) is said to be compatible with the symplectic

fibration π : M → B over the symplectic manifold (F, σ) if for every b ∈ B, the

restriction of ω to Fb agrees with σb . In other words, the maps φα (b)−1 : (F, σ) →

(M, ω) are symplectic embeddings for all α and all b ∈ Uα .

A natural question that arises from here is under which condition the total space

M admits a symplectic form that is compatible with the symplectic fibration. Let

us review the Kodaira-Thurston Example 5.1. We can see M as R4 /Γ where Γ is

the discrete group generated by:

g1

g2

g3

g4

: (x1 , x2 , y1 , y2 ) 7→(x1 + 1, x2 , y1 , y2 )

: (x1 , x2 , y1 , y2 ) 7→(x1 , x2 + 1, y1 + y2 , y2 )

: (x1 , x2 , y1 , y2 ) 7→(x1 , x2 , y1 + 1, y2 )

: (x1 , x2 , y1 , y2 ) 7→(x1 , x2 , y1 , y2 + 1)

The gi all verify gi∗ ω0 = dx1 ∧ dx2 + dy1 ∧ dy2 , so Γ ⊂ Symp(R4 , ω0 ). But one can

notice that M is also a symplectic T2 -bundle, with base manifold T 2 (given by the

first two components of (x1 , x2 )) and fiber T˜2 (given by the last two components

(y1 , y2 )). More precisely, we have the following fibration diagram:

IdM

M

T 2 × T˜2

π

pr(x1 ,x2 )

T

2

From this point of view, however, there is no guarantee for M to be a symplectic

manifold. The following result, due to Thurston, is a powerful tool to create symplectic manifolds in the context of symplectic fibrations. Further, it allows us to

conclude that the Kodaira-Thurston example is a symplectic manifold under this

new approach (we will discuss this application after the proof).

Theorem 6.3 (Thurston). Let π : M → B be a compact symplectic fibration with

symplectic fibre (F, σ) and connected symplectic base (B, β). Denote by σb ∈ Ω2 (Fb )

12

the canonical symplectic form on the fibre Fb and suppose that there is a cohomology

class a ∈ H 2 (M ) such that

i∗b a = [σb ]

for all b ∈ B (for some b ∈ B is sufficient). Then, for every sufficiently large real

number K > 0, there exists a symplectic form ωK ∈ Ω2 (M ) which is compatible with

the fibration π and represents the class a + K[π ∗ β].

Proof. Let τ0 ∈ Ω2 (M ) be any closed 2-form which represents the class a ∈ H 2 (M ).

For any α, denote by σα ∈ Ω2 (Uα × F ) the 2-form obtained from σ via the pullback

under the projection Uα × F → F . Since σ is closed, it follows from the definition

of σα that φ∗α σα is closed. If the open sets Uα ⊂ B in the cover are chosen to be

contractible, then ι∗b φ∗α σα = σb . Since ι∗b a = [σb ] it follows that φ∗α σα is a representation of the class a. Because τ0 is also a representation of a, it follows that

[φ∗α σα − τ0 ] = [0]. Hence, the 2-forms φ∗α σα − τ0 ∈ Ω2 (π −1 (Uα )) are exact.

Choose a collection of 1-forms λα ∈ Ω1 (π −1 (Uα )) such that φ∗α σα − τ0 = d λα . Now

choose a partition of unity ρα : B P

→ [0, 1] which is subordinate to the cover {Uα }α

and define τP

∈ Ω2 (M ) by τ := τ0 + α d((ρα ◦π)λα ). This 2-form is P

closed, since τ0 is

closed, and α d((ρα ◦ π)λα ) is exact. Because τ0 represents a and α d((ρα ◦ π)λα )

is exact, τ represents a. Moreover, the 1-form d(ρα ◦ π) vanishes on vectors tangent

to the fiber and hence

X

ι∗b τ = ι∗b τ0 +

((ρα ◦ π)ι∗b d λα )

α

=

X

=

X

=

X

=

X

(ρα ◦ π)ι∗b (τ0 + d λα )

α

(ρα ◦ π)ι∗b (τ0 + φ∗α σα − τ0 )

α

(ρα ◦ π)ι∗b φ∗α σα

α

(ρα ◦ π)σb

α

= σb

The form τ need not be nondegenerate. However, it is nondegenerate on the (vertical) subspaces

Vertx = ker d π(x) ⊂ Tx M.

Indeed, let v, w ∈ Vertx , then we have τ (v, w) = τ0 (v, w), and since σb is nondegenerate for all b it follows from the condition on a that τ0 is also nondegenerate, so τ

is nondegenerate on Vertx .

Hence τ determines a field of horizontal subspaces

Horx = (Vertx )τ = {ξ ∈ Tx M : τ (ξ, η) = 0 ∀η ∈ Vertx }.

The subspace Horx ⊂ Tx M is a horizontal complement of Vertx and the projection

dπ(x) : Horx → Tπ(x) B is a bijection.

Now the form ωK = τ + Kπ ∗ β agrees with τ on Vertx (since Kπ ∗ β = 0 on Vertx ).

Moreover, ωK is nondegenerate on Horx for K > 0 sufficiently large, since π ∗ β is

nondegenerate on Horx . Lastly, ωK (ξ, η) = 0 for ξ ∈ Horx and η ∈ Vertx (since

τ (ξ, η) = 0 by the definiton of Horx , and π ∗ β(ξ, η) = 0 because η ∈ ker dπ(x)).

Hence, ωK is nondegenerate for K sufficiently large.

13

Assume now that π : M → B is a locally trivial fibration and that the fiber F

is orientable of dimension 2. From Example 2.8 we know that we have a symplectic

form ωF and the structure group always reduce to Symp(F, ωF ). Hence, we can turn

the fibration π : M → B into a symplectic fibration. This is the case in the KodairaThurston example, which also satisfies the cohomological assumption of Theorem

6.3. This procedure can be used to construct many more examples.

7

Fibre Connected Sum

We aim to define the fibre connected sum and state an interesting property about

the fundamental group (Lemma 7.5), which will be necessary for the construction

in the proof of Gompf’s Theorem.

Given a symplectic submanifold Q of M , we denote N (Q) any sufficiently small

open neighbourhood of Q. First of all we need the following result, that can be

proven applying Tubular Neighbourhood and Moser Isotopy theorems.

Theorem 7.1 (Symplectic Neighbourhood). Let j = 1, 2. Consider (Mj , ωj ) symplectic manifolds with compact symplectic submanifolds Qj . Assume there is an

isomorphism of the symplectic normal bundles Φ : νQ1 → νQ2 which covers the symplectomorphism ϕ : (Q1 , ω1 ) → (Q2 , ω2 ). Then ϕ extends to a symplectomorphism

between neighbourhoods ψ : (N (Q1 ), ω1 ) → (N (Q2 ), ω2 ) such that d ψ induces Φ via

⊥

νQ j ∼

= T Q⊥

j ⊂ T N (Qj ) (where T Qj is defined using ωj ).

We use the above theorem to define the fibre connected sum. For j = 1, 2,

let (Mj , ωj ) be two symplectic manifolds of dimension 2n. Consider a symplectic

manifold (Q, τ ) of dimension 2n − 2 for which there exist symplectic embeddings

ij : Q → Mj and Qj := ij (Q) has trivial normal bundle for j = 1, 2.

Let N (Qj ) be neighbourhoods such that Theorem 7.1 holds. Then we can choose

a sufficiently small for which there is a map fj : Q×B() → N (Qj ) ⊆ Mj satisfying

fj∗ ωj = τ × dx ∧ dy and fj (q, 0) = ij (q) for any q ∈ Q and for both j = 1, 2, where

B() denotes the 2-dimensional open ball of radius . For some 0 < δ < , we choose

a symplectomorphism

φ : B() − B(δ) → B() − B(δ)

that interchanges the two boundaries. We define Nδ (Qj ) := fj (Q × B(δ)) ⊂ N (Qj ).

Definition 7.2. The fiber connected sum of M1 and M2 among U is defined as

M1 #Q M2 := M1 − Nδ (Q1 ) ∪φ M2 − Nδ (Q2 )

The notation ∪φ means that we are identifying the elements that are equal under

φ, i.e. in M1 #Q M2 we have

f2 (q, r) = f1 (q, φ(r))

∀q ∈ Q, ∀r ∈ B() − B(δ)

With this definition, since φ is a symplectomorphism, the two forms ω1 , ω2 agree

on the overlap, giving a symplectic structure on M1 #Q M2 .

Remark 7.3. The definition of M1 #Q M2 depends on the choice of fj , and δ. In

particular, different fj can give rise to different manifolds M1 #Q M2 .

14

Let us consider a surface we will need to state Lemma 7.5.

Example 7.4. Blow-up techniques, which we will not discuss in detail, allow us

to consider manifolds with interesting properties. This is the case of the following

surface.

Consider two transverse nonsingular cubics Γ1 and Γ2 in CP 2 , their intersection

has exactly 9 points: Γ1 ∩ Γ2 = {x1 , . . . , x9 }. Blowing up CP 2 at x1 , . . . , x9 gives us

a 4-manifold V that contains an embedded torus T which has trivial normal bundle

and such that V − T is simply connected. As a matter of fact, this is the only

property we need to know about V in order to prove Lemma 7.5.

We can finally state the following result. Its proof uses multiple instances of

van Kampen’s Theorem and has many technicalities that are left to the interested

reader.

Lemma 7.5. Consider V and T as in Example 7.4. Let (X, ω) be a symplectic

manifold of the same dimension as V such that there is a symplectic torus T 0 with

trivial normal bundle. Rescaling the form on V such that T 0 and T have the same

area, we define X 0 := X#T 0 V . Then

π1 (X 0 ) =

π1 (X)

hi∗ (π1 (T 0 ))i

where hi∗ (π1 (T 0 ))i is the normal subgroup generated by the image of π1 (T 0 ) via the

inclusion i : T 0 → X.

8

Gompf’s Theorem

In this section we will prove the following deep result, which will need several nontrivial reasonings as well as Example 7.4 and Lemma 7.5.

Theorem 8.1 (Gompf). Let G be a finitely presented group. Then there exists a

compact symplectic 4-manifold (i.e. a manifold of real dimension 4) with fundamental group G.

In particular, the proof of the theorem will give an explicit construction of infinitely many compact symplectic 4-manifolds that are not Kähler. Indeed, it suffices

to choose G such that its abelianization will be Zk with k odd: under this condition, Proposition 4.14 immediately entails that the considered 4-manifold cannot be

Kähler.

Let us prove the theorem by explicit construction. Let G be presented by generators gi with 1 ≤ i ≤ k and relations rj (g1 , . . . , gk ) = 1G with 1 ≤ j ≤ ` for

some k, ` ∈ N, so that we can write G = hg1 , . . . , gk | r1 , . . . , r` i. Let (F, ω1 ) be a

compact oriented Riemann surface of genus k, i.e. a connected sum of k tori. We

can choose k oriented simple closed curves α1 , . . . , αk and k similar curves β1 , . . . , βk

representing the basis of H1 (F ), such that the αi go “around” the k holes of F and

the βi go “through” the k holes; in particular, we require that αi · βj = δij . In other

words, if the 1-forms σiα are the orientation 1-forms of the αi and the σjβ are the

orientations of the βj , then each αi only intersects βi and only in one point pi , such

that σiα ∧ σiβ forms a positive basis for Tp∗i F .

15

Interpreting αi and βj as loops, we have

π1 (F ) = hα1 , . . . , αr , β1 , . . . , βr | α1 β1 α1−1 β1−1 · · · αr βr αr−1 βr−1 = 1i,

so that π(F )/hβ1 , . . . , βk i is the free group generated by the classes of the αi . For 1 ≤

i ≤ `, the relation ri ([α1 ], . . . , [αk ]) is a well-defined element and can be represented

by a curve γi in F . This curve is closed but not necessarily simple.

Setting γ`+j = βj , we get G ∼

= π1 (F )/hγ1 , . . . , γ`+k i. Our goal is to get rid of the

γi by symplectically glueing a simply-connected space along them. First, we need

to define a closed 1-form on F . This can be done by changing a little the surface F ,

as precised in the following.

Lemma 8.2. Up to adding tori to F , there exists a closed 1-form ρ on F which

restricts to a volume form on each γi .

Proof. Without loss of generality, assume the γi to be in general position, S

that

is when they intersect transversally in pairs. Under this assumption, Γ := i γi

becomes an oriented graph on F , where each edge is given by a part of some γi

between the intersection with other paths.

1. Choose two distinct points x, y ∈ S 1 and consider the paths

α := S 1 × x,

β := x × S 1 ,

γ := y × S 1

in T2 := S 1 ×S 1 . Next, we take p ∈ γ and a small disk D 3 p such that D∩α =

∅ = D ∩ β and D ∩ γ is connected (see Figure 2 for a better understanding).

p

D

γ

α

β

Figure 2: Torus, notation as in item 1 of proof of Lemma 8.2.

2. With the notationR of 1, we want to prove that there exists a 1-form ν such

that ν|D = 0 and ν > 0 over α, β and γ.

Since T2 = α × β, we can pick the 1-forms dual to α and β, that we call θ and

φ respectively, and define

δ := πα∗ θ + πβ∗ φ

where πα and πβ are the projections from the torus. Since θ and φ are closed,

so is δ, hence there exists a smooth function g such that δ|D = d g. Extending

g to g˜ smoothly outside D such that g˜ quickly goes to 0, we finally define

ν := δ − d g˜, which behaves as required.

16

3. Now we want to modify F such that it is possible to define a 1-form whose

integral is positive on each edge of Γ. In order to do this, we will use 2.

Formally, for each edge e, pick pe ∈ e and a small disk De 3 pe that does not

intersect any edge except e, and such that De ∩ e is connected (hence simply

connected by the topology on e). We glue a torus to F to get the connected

sum F #T2 , where we identify D ∼

= De such that γ ∩ D is sent to e ∩ De

(notation as in 1). Now (e \ De ) ∪ (γ \ D) is an edge (that can be made smooth

by reparametrising) in the new surface F #T2 ; we redefine e to be this new

edge.

4. With another abuse of notation, we call the obtained permuted paths γi , and

the resulting surface F . The genus of this new F has grown, so we additionally

need to consider paths γ`+k+1 , . . . , γm such that π1 (F )/hγ1 , . . . , γm i ∼

= G (these

paths will correspond to α and β in T2 ).

5. Let ν as in item 2. Set ρ∗ to be the closed 1-form on F that equals ν on T2 − D

for any torus added

in step 3, and 0 otherwise. By construction, for any edge

R

e ∈ Γ, we get e ρ∗ > 0.R Therefore

a volume

R ∗ for any i = 1, . . . , m there exists

R

form θi on γi such that e θi = e ρ for all edges e ⊆ γi . Hence e θi − ρ∗|γi = 0,

i.e. θi − ρ∗|γi = d fi for some smooth fi . Further, we choose fi such that it

equals zero on each vertex of Γ in γi . Perturbing θi , we can moreover require

fi = 0 near each vertex. The function f := f1 + · · · + fm is then smooth and

can be extended smoothly on F . Finally, let ρ := ρ∗ + d f .

This ρ is a closed 1-form which restricts to the volume form θi on γi , as required.

It is slightly troublesome that we had to add tori to F , but we will see that the

influence of the new γi (i.e. with ` + k + 1 ≤ i ≤ m, as defined in item 4 of the

proof) will not change the argument of the proof.

In order to get rid of the γi (now for 1 ≤ i ≤ m), so that the fundamental group

becomes G, we would need some more dimensions (that is the reason why in the

statement we talk about a 4-manifold). Let (T2 , ω2 ) be the symplectic torus and

define X := F × T2 with symplectic form ω := ω1 × ω2 , and let α be the oriented

simple closed curve S 1 × x ⊂ T2 for some x ∈ S 1 . Let θ be a 1-form on T2 that is

positive on positively oriented tangent vectors of α, so if σα is the orientation 1-form

of α, then θ restricts to f · σα for some function f > 0.

Define Ti := γi × α ⊂ X. These spaces are immersed tori (embedded if γi is

embedded) and ρ ∧ θ is (strictly) positive on the positively oriented bases of each Ti ,

hence (Ti , ρ ∧ θ) is a symplectic manifold for 1 ≤ i ≤ m. Moreover, ω1 and ω2 both

vanish on each T Ti (since they need two tangent vectors in F respectively T2 ), so ω

vanishes on the tangent bundle of each Ti .

Define ω 0 := ω + ρ ∧ θ. This is a globally well-defined 2-form, and restricts to

ρ ∧ θ on the Ti . Since (ω 0 )2 = ω 2 + (ρ ∧ θ)2 + ω ∧ (ρ ∧ θ) + (ρ ∧ θ) ∧ ω = ω 2 is

non-vanishing, ω 0 is nondegenerate, hence again a symplectic form.

Now the space (X, ω 0 ) is symplectic, and has symplectically immersed tori (Ti , ω 0 ).

Writing X = F × T2 = F × S 1 × α and noting that F × S 1 is three dimensional,

we can perturb the γi in F × S 1 to make them embedded and disjoint, giving us

symplectically embedded tori Ti .

We want to glue copies of the elliptic surface V defined in Example 7.4 along

these tori, which will make the γi and α contractible. We will also need to make

17

sure π1 (T2 ) does not contribute, so we will also glue a copy along {z} × T2 for z ∈ F

such that z does not belong to any γi . We just have to make sure we may actually

glue V along these tori, i.e. we need to prove the following.

Lemma 8.3. The embedded tori (Ti , ω 0 ) and ({z} × T2 , ω 0 ) of (X, ω 0 ) have trivial

normal bundle.

Proof. The statement is clear for ({z} × T2 , ω 0 ) since for p ∈ T2 , T(z,p) X splits into

Tz F and Tp T2 , hence the normal bundle is (Tz F ) × T2 → T2 which is trivial.

For (Ti , ω 0 ), note that the normal bundle is the pull-back of the normal bundle

of γi ⊂ F × S 1 , which is the pullback of normal bundle of γi ⊂ F before perturbing.

Since F is an oriented surface and γi an oriented curve, the normal bundle cannot

be the Möbius strip and hence must be trivial. Any pullback of a trivial bundle is

again trivial, so we find that (Ti , ω 0 ) has trivial normal bundle.

Finally, we are able to glue copies of V along the Ti and along {z} × T2 , and call

e Applying Lemma 7.5 for each gluing, we get that

the resulting space X.

e = π1 (F )/hγ1 , . . . , γm i ∼

π1 (X)

= G.

This concludes the proof of Theorem 8.1.

9

Discussion on McDuff example

Eventually we showed that there is a large range of examples of symplectic non

Kähler manifolds. However, the previous examples rely on obstructions on the rank

of the fundamental group. McDuff provided the first simply connected symplectic

non Kähler manifold [5] using blow-up techniques. This process uses the Gromov

embedding theorem.

The strong Whitney embedding theorem states that any n-dimensional smooth

manifold can be smoothly embedded in R2n . In the same fashion, one can wonder

if we can symplectically embed compact manifolds into a universal one as well. A

first idea would be to consider (R2n , ω0 ). However, there is a natural obstruction

to this: ω0 is exact, so the cohomology class of its pull back has to vanish as

well, but a symplectic form on a compact manifold is never exact because its top

power is a volume form. A positive result has been proved by Gromov considering

the projective space: if (M, ω), of real dimension 2n, is endowed with an integral

symplectic form (i.e. [ω] ∈ H 2 (M ; Z)), then it can be symplectically embedded into

CP 2n+1 with its standard form. Gromov also outlined that if we have a symplectic

manifold (N, ωN ) symplectically embedded in (M, ωM ), then one can blow up M

along N in order to form a new symplectic manifold.

The idea presented in [5] is then to consider the manifold M of Thurston-Kodaira

(Example 5.1). As M is 4-dimensional, it can be symplectically embedded into CP 5 ,

˜ w)

and hence we can blow up CP 5 along M . If we denote by (X,

˜ the obtained

symplectic manifold, the result established by McDuff is the following.

˜ w)

Theorem 9.1. The space (X,

˜ is a simply-connected symplectic closed manifold

˜

˜

such that b3 (X) = 3. Hence X is not Kähler.

18

References

[1] R. E. Gompf. “A new construction of symplectic manifolds”. In: Annals of Mathematics (1995), pp. 527–595.

[2] H. Guggenheimer. “Sur les variétés qui possedent une forme extérieure quadratique fermée”. In: Comptes rendus hebdomadaires des séances de l’Académie des

Sciences 232.6 (1951), pp. 470–472.

[3] H. Guggenheimer. Variétés symplectiques. Colloq. Topologie de Strasbourg,

1951.

[4] D. Huybrechts. Complex geometry: an introduction. Springer Science & Business

Media, 2006.

[5] D. McDuff. “Examples of simply-connected symplectic non-Kählerian manifolds”. In: Journal of Differential Geometry 20.1 (1984), pp. 267–277.

[6] D. McDuff and D. Salamon. Introduction to symplectic topology. Third Edition.

Oxford University Press, 2017.

[7] W.P. Thurston. “Some Simple Examples of Symplectic Manifolds”. In: Proceedings of the American Mathematical Society 55.2 (1976), pp. 467–468.

19

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