2Complex structures
III Symplectic Geometry
2.4 Hodge theory
So what we have got so far is that if we have a complex manifold, then we can
decompose
Ω
k
(M; C) =
M
p+q=k
Ω
p,q
,
and using
¯
∂ : Ω
p,q
→ Ω
p,q+1
, we defined the Dolbeault cohomology groups
H
p,q
Dolb
(M) =
ker
¯
∂
im
¯
∂
.
It would be nice if we also had a decomposition
H
k
dR
∼
=
M
p+q=k
H
p,q
Dolb
(M).
This is not always true. However, it is true for compact K¨ahler manifolds:
Theorem
(Hodge decomposition theorem)
.
Let (
M, ω
) be a compact K
¨
haler
manifold. Then
H
k
dR
∼
=
M
p+q=k
H
p,q
Dolb
(M).
To prove the theorem, we will first need a “real” analogue of the theorem.
This is an analytic theorem that lets us find canonical representatives for each
cohomology class. We can develop the same theory for Dolbeault cohomology,
and see that the canonical representatives for Dolbeault cohomology are the
same as those for de Rham cohomology. We will not prove the analytic theorems,
but just say how these things piece together.
Real Hodge theory
Let
V
be a real oriented vector space
m
with inner product
G
. Then this induces
an inner product on each Λ
k
=
V
k
(V ), denoted h·, ·i, defined by
hv
1
∧ ··· ∧ v
k
, w
1
∧ ··· ∧ w
k
i = det(G(v
i
, w
j
))
i,j
Let e
1
, . . . , e
n
be an oriented orthonormal basis for V . Then
{e
j
1
∧ ··· ∧ e
j
k
: 1 ≤ j
1
< ··· < j
k
≤ m}
is an orthonormal basis for Λ
k
.
Definition
(Hodge star)
.
The Hodge
∗
-operator
∗
: Λ
k
→
Λ
m−k
is defined by
the relation
α ∧ ∗β = hα, βi e
1
∧ ··· ∧ e
m
.
It is easy to see that the Hodge star is in fact an isomorphism. It is also not
hard to verify the following properties:
Proposition.
– ∗(e
1
∧ ··· ∧ e
k
) = e
k+1
∧ ··· ∧ e
m
– ∗(e
k+1
∧ ··· ∧ e
m
) = (−1)
k(m−k)
e
1
∧ ··· ∧ e
k
.
– ∗∗ = α = (−1)
k(m−k)
α for α ∈ Λ
k
.
In general, let (
M, g
) be a compact real oriented Riemannian manifold of
dimension
m
. Then
g
gives an isomorphism
T M
∼
=
T
∗
M
, and induces an inner
product on each
T
∗
p
M
, which we will still denote
g
p
. This induces an inner
product on each
V
k
T
∗
p
M, which we will denote h·, ·i again.
The Riemannian metric and orientation gives us a volume form
Vol ∈
Ω
m
(
M
),
defined locally by
Vol
p
(e
1
∧ ··· ∧ e
m
),
where
e
1
, . . . , e
m
is an oriented basis of
T
∗
p
M
. This induces an
L
2
-inner product
on Ω
k
(M),
hα, βi
L
2
=
Z
M
hα, βi Vol.
Now apply Hodge
∗
-operator to each (
V, G
) = (
T
∗
p
M, g
p
) and
p ∈ M
. We then
get
Definition
(Hodge star operator)
.
The Hodge
∗
-operator on forms
∗
: Ω
k
(
M
)
→
Ω
m−k
(M) is defined by the equation
α ∧ (∗β) = hα, βi Vol.
We again have some immediate properties.
Proposition.
(i) ∗ ∗ α = (−1)
k(m−k)
α for α ∈ Ω
k
(M).
(ii) ∗1 = Vol
We now introduce the codifferential operator
Definition
(Codifferential operator)
.
We define the codifferential operator
δ
:
Ω
k
→ Ω
k−1
to be the L
2
-formal adjoint of d. In other words, we require
hdα, βi
L
2
= hα, δβi
L
2
for all α ∈ Ω
k
and β ∈ Ω
k+1
.
We immediately see that
Proposition. δ
2
= 0.
Using the Hodge star, there is a very explicit formula for the codifferential
(which in particular shows that it exists).
Proposition.
δ = (−1)
m(k+1)+1
∗d ∗ : Ω
k
→ Ω
k−1
.
Proof.
hdα, βi
L
2
=
Z
M
dα ∧ ∗β
=
Z
M
d(α ∧ ∗β) − (−1)
k
Z
M
α ∧ d(∗β)
= (−1)
k+1
Z
M
α ∧ d(∗β) (Stokes’)
= (−1)
k+1
Z
M
(−1)
(m−k)k
α ∧ ∗ ∗ d(∗β)
= (−1)
k+1+(m−k)k
Z
M
hα, ∗ d ∗βi.
We can now define the Laplace–Beltrami operator
Definition
(Laplace–Beltrami operator)
.
We define the Laplacian, or the
Laplace–Beltrami operator to be
∆ = dδ + δd : Ω
k
→ Ω
k
.
Example.
If
M
=
R
m
with the Euclidean inner product, and
f ∈
Ω
0
(
M
) =
C
∞
(M), then
∆f = −
n
X
i=1
∂
2
f
∂x
2
i
.
It is an exercise to prove the following
Proposition.
(i) ∆∗ = ∗∆ : Ω
k
→ Ω
m−k
(ii) ∆ = (d + δ)
2
(iii) h∆α, βi
L
2
= hα, ∆βi
L
2
.
(iv) ∆α = 0 iff dα = δα = 0.
In particular, (iv) follows from the fact that
h∆α, αi = hdα, dαi + hδα, δαi = kdαk
2
L
2
+ kδαk
2
L
2
.
Similar to IA Vector Calculus, we can define
Definition (Harmonic form). A form α is harmonic if ∆α = 0. We write
H
k
= {α ∈ Ω
k
(m) | ∆α = 0}
for the space of harmonic forms.
Observe there is a natural map
H
k
→ H
k
dR
(
M
), sending
α
to [
α
]. The main
result is that
Theorem
(Hodge decomposition theorem)
.
Let (
M, g
) be a compact oriented
Riemannian manifold. Then every cohomology class in
H
k
dR
(
M
) has a unique
harmonic representation, i.e. the natural map
H
k
→ H
k
dR
(
M
) is an isomorphism.
We will not prove this.
Complex Hodge theory
We now see what we can do for complex K¨ahler manifolds. First check that
Proposition.
Let
M
be a complex manifold,
dim
C
M
=
n
and (
M, ω
) K¨ahler.
Then
(i) ∗ : Ω
p,q
→ Ω
n−p,n−q
.
(ii) ∆ : Ω
p,q
→ Ω
p,q
.
define the
L
2
-adjoints
¯
∂
∗
=
± ∗
¯
∂∗
and
∂
∗
=
− − ∗∂∗
with the appropriate
signs as before, and then
d = ∂ +
¯
∂, δ = ∂
∗
+
¯
∂
∗
.
We can then define
∆
∂
= ∂∂
∗
+ ∂
∗
∂ : Ω
p,q
→ Ω
p,q
∆
¯
∂
=
¯
∂
¯
∂
∗
+
¯
∂
∗
¯
∂ : Ω
p,q
→ Ω
p,q
.
Proposition. If our manifold is K¨ahler, then
∆ = 2∆
∂
= 2∆
¯
∂
.
So if we have a harmonic form, then it is in fact
∂
and
¯
∂
-harmonic, and in
particular it is ∂ and
¯
∂-closed. This give us a Hodge decomposition
H
k
C
=
M
p+q=k
H
p,q
,
where
H
p,q
= {α ∈ Ω
p,q
(M) : ∆α = 0}.
Theorem
(Hodge decomposition theorem)
.
Let (
M, ω
) be a compact K¨ahler
manifold. The natural map H
p,q
→ H
p,q
Dolb
is an isomorphism. Hence
H
k
dR
(M; C)
∼
=
H
k
C
=
M
p+q=k
H
p,q
∼
=
M
p+q=k
H
p,q
Dolb
(M).
What are some topological consequences of this? Recall the Betti numbers
are defined by
b
k
= dim
R
H
k
dR
(M) = dim
C
H
k
dR
(M; C).
We can further define the Hodge numbers
h
p,q
= dim
C
H
p,q
Dolb
(M).
Then the Hodge theorem says
b
k
=
X
p+q=k
h
p,q
.
Moreover, since
H
p,q
Dolb
(M) = H
q,p
Dolb
(M).
So we have Hodge symmetry,
h
p,q
= h
q,p
.
Moreover, the ∗ operator induces an isomorphism
H
p,q
Dolb
∼
=
H
n−p,n−q
Dolb
.
So we have
h
p,q
= h
n−p,n−q
.
There is called central symmetry, or Serre duality. Thus, we know that
Corollary. Odd Betti numbers are even.
Proof.
b
2k+1
=
X
p+q=2k+1
h
p,q
= 2
k
X
p=0
h
p,2k+1−p
!
.
Corollary. h
1,0
= h
0,1
=
1
2
b
1
is a topological invariant.
We have also previously seen that for a general compact K¨ahler manifold, we
have
Proposition. Even Betti numbers are positive.
Recall that we proved this by arguing that [
ω
k
]
6
= 0
∈ H
2k
(
M
). In fact,
ω
k
∈ H
k,k
Dolb
(M), and so
Proposition. h
k,k
6= 0.
We usually organize these h
p,q
in the form of a Hodge diamond, e.g.
h
0,0
h
1,0
h
0,1
h
2,0
h
1,1
h
0,2
h
2,1
h
1,2
h
2,2