Examples of Symplectic Manifolds (22) (PDF)

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