COMPLEX GEOMETRY Course notes

5.2 Some applications of the Main Theorem
Theorem 5.2.1 (Riemannian case). Let H k be the space of harmonic k-forms. Then the map
H k −→ HdR k (X, R (or C)) given by α ↦→ [α], which is well defined since dα = 0, is an isomorphism.
Proof: By the Main Theorem, β ∈ A k can be written as α + ∆γ = α + dd ∗ γ + d ∗ dγ, where α +
dd ∗ α is d-closed. Since β is closed we have d ∗ dγ is closed and hence it belongs to (Im(d ∗ )) ⊥ . So
0 = (dγ, dd ∗ dγ) = ||d ∗ dγ|| 2 . Similarly, we get the analogous theorem for ∆ ∂E
where we can identify
H q (X, O(E) = E) with H 0,q (X, E) via the Dolbeaut isomorphism.
∂
Theorem 5.2.2. Given a holomorphic vector bundle E −→ X, let H 0,q (E) be the space of (∂ E )-holomorphic
forms of type (0, q) with values in E. Then the map H 0,q (E) −→ H q (X, E) given by α ↦→ [α] is an isomorphism.
Corollary 5.2.1. If X be a compact manifold then H q (X, R) is finite dimensional. If X is a compact
complex manifold and E −→ X is a holomorphic vector bundle then H q (X, E) is finite dimensional.
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