2Complex structures
III Symplectic Geometry
2.1 Almost complex structures
Symplectic manifolds are very closely related to complex manifolds. A first
(weak) hint of this fact is that they both have to be even dimensional! In general,
symplectic manifolds will have almost complex structures, but this need not
actually come from a genuine complex structure. If it does, then we say it is
K¨ahler, which is a very strong structure on the manifold.
We begin by explaining what almost complex structures are, again starting
from the linear algebraic side of the story. A complex vector space can be thought
of as a real vector space with a linear endomorphism that acts as “multiplication
by i”.
Definition
(Complex structure)
.
Let
V
be a vector space. A complex structure
is a linear J : V → V with J
2
= −1.
Here we call it a complex structure. When we move on to manifolds, we will
call this “almost complex”, and a complex structure is a stronger condition. It
is clear that
Lemma.
There is a correspondence between real vector spaces with a complex
structure and complex vector spaces, where J acts as multiplication by i.
Our symplectic manifolds come with symplectic forms on the tangent space.
We require the following compatibility condition:
Definition
(Compatible complex structure)
.
Let (
V,
Ω) be a symplectic vector
space, and
J
a complex structure on
V
. We say
J
is compatible with Ω if
G
J
(u, v) = Ω(u, Jv) is an inner product. In other words, we need
Ω(Ju, Jv) = Ω(u, v), Ω(v, Jv) ≥ 0
with equality iff v = 0.
Example. On the standard symplectic vector space (R
2n
, Ω), we set
J
0
(e
i
) = f
i
, J
0
(f
i
) = −e
i
.
We can then check that this is compatible with the symplectic structure, and in
fact gives the standard inner product.
Proposition
(Polar decomposition)
.
Let (
V,
Ω) be a symplectic vector space,
and
G
an inner product on
V
. Then from
G
, we can canonically construct a
compatible complex structure
J
on (
V,
Ω). If
G
=
G
J
for some
J
, then this
process returns J.
Note that in general, G
J
(u, v) = Ω(u, Jv) 6= G(u, v).
Proof. Since Ω, G are non-degenerate, we know
Ω(u, v) = G(u, Av)
for some
A
:
V → V
. If
A
2
=
−
1, then we are done, and set
J
=
A
. In general,
we also know that
A
is skew-symmetric, i.e.
G
(
Au, v
) =
G
(
u, −Av
), which is
clear since Ω is anti-symmetric. Since
AA
t
is symmetric and positive definite, it
makes sense to write down
√
AA
T
(e.g. by diagonalizing), and we take
J =
√
AA
t
−1
A =
p
−A
2
−1
A.
It is clear that
J
2
=
−
1, since
A
commutes with
√
AA
t
, so this is a complex
structure. We can write this as
A
=
√
AA
T
J
, and this is called the (left) polar
decomposition of A.
We now check that
J
is a compatible, i.e.
G
J
(
u, v
) = Ω(
u, Jv
) is symmetric
and positive definite. But
G
J
(u, v) = G(u,
√
AA
t
v),
and we are done since
√
AA
t
is positive and symmetric.
Notation.
Let (
V,
Ω) be a symplectic vector space. We write
J
(
V,
Ω) for the
space of all compatible complex structures on (V, Ω).
Proposition. J(V, Ω) is path-connected.
Proof.
Let
J
0
, J
1
∈ J
(
V,
Ω). Then this induces inner products
G
J
0
, G
J
1
. Let
G
t
= (1
−t
)
G
J
0
+
tG
J
1
be a smooth family of inner products on
V
. Then apply
polar decomposition to get a family of complex structures that start from
J
0
to
J
1
.
A quick adaptation of the proof shows it is in fact contractible.
We now move on to the case of manifolds.
Definition
(Almost complex structure)
.
An almost complex structure
J
on
a manifold is a smooth field of complex structures on the tangent space
J
p
:
T
p
M → T
p
M, J
2
p
= −1.
Example.
Suppose
M
is a complex manifold with local complex coordinates
z
1
, . . . , z
n
on U ⊆ M. We have real coordinates {x
i
, y
i
} given by
z
i
= x
i
+ iy
i
.
The tangent space is spanned by
∂
∂x
i
,
∂
∂y
i
. We define J by
J
p
∂
∂x
j
=
∂
∂y
j
, J
p
∂
∂y
j
= −
∂
∂x
j
.
The Cauchy–Riemann equations imply this is globally well-defined. This
J
is
called the canonical almost-complex structure on the complex manifold M.
Definition
(Integrable almost complex structure)
.
An almost complex structure
on M is called integrable if it is induced by a complex structure.
Example.
It is a fact that
CP
2
#
CP
2
#
CP
2
has an almost complex but no
complex structure.
Definition
(Compatible almost complex structure)
.
An almost complex struc-
ture
J
on
M
is compatible with a symplectic structure
ω
if
J
p
is compatible with
ω
p
for all p ∈ M. In this case, (ω, g
J
, J) is called a compatible triple.
Any two of the structures determine the three, and this gives rise to the nice
fact that the intersection of any two of O(2n), Sp(2n, R) and GL(n, C) is U(n).
Performing polar decomposition pointwise gives
Proposition.
Let (
M, ω
) be a symplectic manifold, and
g
a metric on
M
. Then
from
g
we can canonically construct a compatible almost complex structure
J.
As before, in general, g
J
( ·, ·) 6= g( ·, ·).
Corollary.
Any symplectic manifold has a compatible almost complex structure.
The converse does not hold. For example,
S
6
is almost complex but not
symplectic.
Notation.
Let (
M, ω
) be a symplectic manifold. We write
J
(
V,
Ω) for the space
of all compatible almost complex structures on (M, ω).
The same proof as before shows that
Proposition. J(M, ω) is contractible.
Proposition.
Let
J
be an almost complex structure on
M
that is compatible
with ω
0
and ω
1
. Then ω
0
and ω
1
are deformation equivalent.
Proof.
Check that
ω
t
= (1
− t
)
ω
0
+
tω
1
works, which is non-degenerate since
ω
t
(
·, J ·
) is a positive linear combination of inner products, hence is non-
degenerate.
Proposition.
Let (
M, ω
) be a symplectic manifold,
J
a compatible almost
complex structure. If
X
is an almost complex submanifold of (
M, J
), i.e.
J(T X) = T X, then X is a symplectic submanifold of (M, ω).
Proof.
We only have to check
ω|
T X
is non-degenerate, but Ω(
·, J ·
) is a metric,
so is in particular non-degenerate.
We shall not prove the following theorem:
Theorem
(Gromov)
.
Let (
M, J
) be an almost complex manifold with
M
open,
i.e.
M
has no closed connected components. Then there exists a symplectic form
ω
in any even 2-cohomology class and such that
J
is homotopic to an almost
complex structure compatible with ω.