COMPLEX GEOMETRY Course notes

5.3 Review
Theorem 5.3.1. Let X be a compact manifold:
{ (⊕
k
(F, P ) =
Ωk X , ∆ )
( d
⊕ )
q Ω0,q (E), ∆ ∂E
if X is Riemannian,
E −→ X is a holomorphic vector bundle
where P is an elliptic or Laplacian-like operator. This implies that
where Ker(P ) is finite dimensional.
C ∞ (F ) = Ker(P )⊥○P (C ∞ (F ))
Corollary 5.3.1.
(1) There is an isomorphism H k ∼
−→ H k dR (X, Ror C) given by α ↦→ [α], where Hk is the finite dimensional
space of harmonic forms.
(2) There is an isomorphism H 0,q ∼
−→ H q (X, E), where H 0,q is the finite dimensional space of (0, q)-forms.
Both isomorphisms depend on the chosen metric.
Note that
HdR k (X, R) ∼= 1
Hk (X, R)
∼= 2 ∼ =
3
Ȟ(H, R)
where ∼ = 1 is given by the de Rham isomorphism, ∼ = 2 and ∼ = 3 are given by the Poincaré Lemma. Recall the
Dolbeaut isomorphism:
(X, E) ∼ = H 0,q (X, ∂ ∂
Ωp E) ∼ = (∗) H p,q (X, Ω p E) =: H p,q (X, E) (finite dimensional),
Ω p E ∼ = O(∆ p TX ∗ ⊗ E) (space of holomorphic p-forms with values in E).
H p,q
where (∗) comes from the isomorphism Ω p E = Ker∂ 0 (A p,0 )(E) −→ A p,1 (E)). Also,
H p,q (X) = H p,q (X, C) = H q (X, Ω p X
) (sheaf of holomorphic functions).
It is known that ∆ = ∂∆ ∂
in every Kähler manifold. Then the following theorem follows:
Theorem 5.3.2. Let X be a Kähler manifold. Then
H k (X, C) =
⊕
H p,q (X).
p+q=k
In other words, if [α] ∈ H k (X, C) with α harmonic, then α = ∑ p+q=k αp,q , where α p,q is the (harmonic)
component of type (p, q).
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