2Complex structures
III Symplectic Geometry
2.3 K¨ahler manifolds
In the best possible scenario, we would have a symplectic manifold with a
compatible, integrable almost complex structure. Such manifolds are called
K¨ahler manifolds.
Definition
(K¨ahler manifold)
.
A K¨ahler manifold is a symplectic manifold
equipped with a compatible integrable almost complex structure. Then
ω
is
called a K¨ahler form.
Let (
M, ω, J
) be a K¨ahler manifold. What can we say about the K¨ahler form
ω? We can decompose it as
ω ∈ Ω
2
(M) ⊆ Ω
2
(M, C) = Ω
2,0
⊕ Ω
1,1
⊕ Ω
2,2
.
We claim that
Lemma. ω ∈ Ω
1,1
.
Proof. Since ω( ·, J ·) is symmetric, we have
J
∗
ω(u, v) = ω(Ju, Jv) = ω(v, JJu) = −ω(v, −u) = ω(u, v).
So J
∗
ω = ω.
On the other hand,
J
∗
acts on holomorphic forms as multiplication by
i
and
anti-holomorphic forms by multiplication by
−
1 (by definition). So it acts on
Ω
2,0
and Ω
0,2
by multiplication by
−
1 (locally Ω
2,0
is spanned by d
z
i
∧
d
z
j
, etc.),
while it fixes Ω
1,1
. So ω must lie in Ω
1,1
.
We can explore what the other conditions on ω tell us. Closedness implies
0 = dω = ∂ω +
¯
∂ω = 0.
So we have ∂ω =
¯
∂ω = 0. So in particular ω ∈ H
1,1
Dolb
(M).
In local coordinates, we can write
ω =
i
2
X
j,k
h
j,k
dz
j
∧ d¯z
k
for some
h
j,k
. The fact that
ω
is real valued, so that
¯ω
=
ω
gives some constraints
on what the h
jk
can be. We compute
¯ω = −
i
2
X
j,k
h
jk
d¯z
j
∧ dz
k
=
i
2
X
j,k
h
jk
dz
k
∧ ¯z
j
.
So we have
h
kj
= h
jk
.
The non-degeneracy condition ω
∧n
6= 0 is equivalent to det h
jk
6= 0, since
ω
n
=
i
2
n
n! det(h
jk
) dz
1
∧ d¯z
1
∧ ··· ∧ dz
n
∧ d¯z
n
.
So h
jk
is a non-singular Hermitian matrix.
Finally, we take into account the compatibility condition
ω
(
v, Jv
)
>
0. If we
write
v =
X
j
a
j
∂
∂z
j
+ b
j
∂
∂¯z
j
,
then we have
Jv = i
X
j
a
j
∂
∂z
j
− b
j
∂
∂¯z
j
.
So we have
ω(v, Jv) =
i
2
X
h
jk
(−ia
j
b
k
− ia
j
b
k
) =
X
h
jk
a
j
b
k
> 0.
So the conclusion is that h
jk
is positive definite.
Thus, the conclusion is
Theorem.
A K¨ahler form
ω
on a complex manifold
M
is a
∂
- and
¯
∂
-closed
form of type (1, 1) which on a local chart is given by
ω =
i
2
X
j,k
h
jk
dz
j
∧ d¯z
k
where at each point, the matrix (h
jk
) is Hermitian and positive definite.
Often, we start with a complex manifold, and want to show that it has a
K¨ahler form. How can we do so? First observe that we have the following
proposition:
Proposition.
Let (
M, ω
) be a complex K¨ahler manifold. If
X ⊆ M
is a complex
submanifold, then (
X, i
∗
ω
) is K¨ahler, and this is called a K¨ahler submanifold.
In particular, if we can construct K¨ahler forms on
C
n
and
CP
n
, then we
have K¨ahler forms for a lot of our favorite complex manifolds, and in particular
complex projective varieties.
But we still have to construct some K¨ahler forms to begin with. To do so,
we use so-called strictly plurisubharmonic functions.
Definition
(Strictly plurisubharmonic (spsh))
.
A function
ρ ∈ C
∞
(
M, R
) is
strictly plurisubharmonic (spsh) if locally,
∂
2
ρ
∂z
j
∂ ¯z
k
is positive definite.
Proposition.
Let
M
be a complex manifold, and
ρ ∈ C
∞
(
M
;
R
) strictly
plurisubharmonic. Then
ω =
i
2
∂
¯
∂ρ
is a K¨ahler form.
We call ρ the K¨ahler potential for ω.
Proof. ω
is indeed a 2-form of type (1, 1). Since
∂
2
=
¯
∂
2
= 0 and
∂
¯
∂
=
−
¯
∂∂
,
we know ∂ω =
¯
∂ω = 0. We also have
ω =
i
2
X
j,k
∂
2
ρ
∂z
j
∂¯z
k
dz
j
∧ d¯z
k
,
and the matrix is Hermitian positive definite by assumption.
Example. If M = C
n
∼
=
R
2n
, we take
ρ(z) = |z|
2
=
X
z
k
¯z
k
.
Then we have
h
jk
=
∂
2
ρ
∂z
j
∂¯z
k
= δ
jk
,
so this is strictly plurisubharmonic. Then
ω =
i
2
X
dz
j
∧ d¯z
k
=
i
2
X
d(x
j
+ iy
j
) ∧ d(x
k
− iy
k
)
=
X
dx
k
∧ dy
k
,
which is the standard symplectic form. So (
C
n
, ω
) is K¨ahler and
ρ
=
|z|
2
is a
(global) K¨ahler potential for ω
0
.
There is a local converse to this result.
Proposition.
Let
M
be a complex manifold,
ω
a closed real-valued (1
,
1)-form
and
p ∈ M
, then there exists a neighbourhood
U
of
p
in
M
and a
ρ ∈ C
∞
(
U, R
)
such that
ω = i∂
¯
∂ρ on U.
Proof. This uses the holomorphic version of the Poincar´e lemma.
When ρ is K¨ahler, such a ρ is called a local K¨ahler potential for ω.
Note that it is not possible to have a global K¨ahler potential on a closed
K¨ahler manifold, because if ω =
i
2
∂
¯
∂ρ, then
ω = d
i
2
¯
∂ρ
is exact, and we know symplectic forms cannot be exact.
Example. Let M = C
n
and
ρ(z) = log(|z|
2
+ 1).
It is an exercise to check that ρ is strictly plurisubharmonic. Then
ω
F S
=
i
2
∂
¯
∂(log(|z
2
| + 1))
is another K¨ahler form on C
n
, called the Fubini–Study form.
The reason this is interesting is that it allows us to put a K¨ahler structure
on CP
n
.
Example.
Let
M
=
CP
n
. Using homogeneous coordinates, this is covered by
the open sets
U
j
= {[z
0
, . . . , z
n
] ∈ CP
n
| z
j
6= 0}.
with the chart given by
ϕ
j
: U
j
→ C
n
[z
0
, . . . , z
n
] 7→
z
0
z
j
, . . . ,
z
j−1
z
j
,
z
j+1
z
j
, . . . ,
z
n
z
j
.
One can check that
ϕ
∗
j
ω
F S
=
ϕ
∗
k
ω
F S
. Thus, these glue to give the Fubini–Study
form on CP
n
, making it a K¨ahler manifold.