2Complex structures
III Symplectic Geometry
2.2 Dolbeault theory
An almost complex structure on a manifold allows us to discuss the notion of
holomorphicity, and this will in turn allow us to stratify our
k
-forms in terms of
“how holomorphic” they are.
Let (
M, J
) be an almost complex manifold. By complexifying
T M
to
T M ⊗C
and then extending J linearly, we can split T M ⊗ C into its ±i eigenspace.
Notation.
We write
T
1,0
for the +
i
eigenspace of
J
and
T
1,0
for the
−i
-
eigenspace of
J
. These are called the
J
-holomorphic tangent vectors and the
J-anti-holomorphic tangent vectors respectively.
We write
T
1,0
for the +
i
eigenspace of
J
, and
T
0,1
the
−i
-eigenspace of
J
.
Then we have a splitting
T M ⊗ C
∼
→ T
1,0
⊕ T
0,1
.
We can explicitly write down the projection maps as
π
1,0
(w) =
1
2
(w − iJw)
π
0,1
(w) =
1
2
(w + iJw).
Example.
On a complex manifold with local complex coordinates (
z
1
, . . . , z
n
),
the holomorphic tangent vectors
T
1,0
is spanned by the
∂
∂z
j
, while
T
0,1
is spanned
by the
∂
∂ ¯z
j
.
Similarly, we can complexify the cotangent bundle, and considering the
±i
eigenspaces of (the dual of) J gives a splitting
(π
1,0
, π
0,1
) : T
∗
M ⊗ C
∼
→ T
1,0
⊕ T
0,1
.
These are the complex linear cotangent vectors and complex anti-linear cotangent
vectors.
Example.
In the complex case,
T
1,0
is spanned by the d
z
j
and
T
0,1
is spanned
by the d¯z
j
.
More generally, we can decompose
V
k
(T
∗
M ⊗ C) =
V
k
(T
1,0
⊕ T
0,1
) =
M
p+q=k
V
`
T
1,0
⊗
V
m
T
0,1
≡
M
p+q=k
Λ
p,q
.
We write
Ω
k
(M, C) = sections of Λ
k
(T
∗
M ⊗ C)
Ω
p,q
(M, C) = sections of Λ
p,q
.
So
Ω
k
(M, C) =
M
p+q=k
Ω
p,q
(M, C).
The sections in Ω
p,q
(M, C) are called forms of type (p, q).
Example. In the complex case, we have, locally,
Λ
p,q
p
= C{dz
I
∧ dz
K
: |I| = `, |K| = m}
and
Ω
p,q
=
X
|I|=p,|K|=q
b
I,K
dz
I
∧ d¯z
K
: b
IK
∈ C
∞
(U, C)
.
As always, we have projections
π
p,q
:
V
k
(T
∗
M ⊗ C) → Λ
p,q
.
Combining the exterior derivative d : Ω
k
(
M, C
)
→
Ω
k−1
(
M, C
) with projections
yield the ∂ and
¯
∂ operators
∂ : Ω
p,q
→ Ω
p+1,q
¯
∂ : Ω
p,q
→ Ω
p,q+1
.
Observe that for functions
f
, we have d
f
=
∂f
+
¯
∂f
, but this is not necessarily
true in general.
Definition
(
J
-holomorphic)
.
We say a function
f
is
J
-holomorphic if
¯
∂f
= 0,
and J-anti-holomorphic if ∂f = 0.
It would be very nice if the sequence
Ω
p,q
Ω
p,q+1
Ω
p,q+2
···
¯
∂
¯
∂
¯
∂
were a chain complex, i.e.
¯
∂
2
= 0. This is not always true. However, this is
true in the case where
J
is integrable. Indeed, if
M
is a complex manifold and
β ∈ Ω
k
(M, C), then in local complex coordinates, we can write
β =
X
p+q=k
X
|I|=p,|K|=q
b
I,K
dz
I
∧ d¯z
K
.
So
dβ =
X
p+q=k
X
|I|=p,|K|=q
db
I,K
∧ dz
I
∧ d¯z
K
=
X
p+q=k
X
|I|=p,|K|=q
(∂ +
¯
∂)b
I,K
∧ dz
I
∧ d¯z
K
= (∂ +
¯
∂)
X
p+q=k
X
|I|=p,|K|=q
b
I,K
dz
I
∧ d¯z
K
Thus, on a complex manifold, d = ∂ +
¯
∂.
Thus, if β ∈ Ω
p,q
, then
0 = d
2
β = ∂
2
β + (∂
¯
∂ +
¯
∂∂)β +
¯
∂
2
β.
Since the three terms are of different types, it follows that
∂
2
=
¯
∂
2
= ∂
¯
∂ +
¯
∂∂ = 0.
In fact, the converse of this computation is also true, which we will not prove:
Theorem (Newlander–Nirenberg). The following are equivalent:
–
¯
∂
2
= 0
– ∂
2
= 0
– d = ∂ +
¯
∂
– J is integrable
– N = 0
where N is the Nijenhuis torsion
N(X, Y ) = [JX, JY ] − J[JX, Y ] − J[X, JY ] −[X, Y ].
When our manifold is complex, we can then consider the cohomology of the
chain complex.
Definition
(Dolbeault cohomology groups)
.
Let (
M, J
) be a manifold with an
integrable almost complex structure. The Dolbeault cohomology groups are the
cohomology groups of the cochain complex
Ω
p,q
Ω
p,q+1
Ω
p,q+2
···
¯
∂
¯
∂
¯
∂
.
Explicitly,
H
p,q
Dolb
(M) =
ker(
¯
∂ : Ω
p,q
→ Ω
p,q+1
)
im(
¯
∂ : Ω
p,q−1
→ Ω
p,q
)
.