COMPLEX GEOMETRY Course notes

Note that HdR 0 (Z) = C if Z is connected. If Z is also compact, by the Poincaré duality Theorem we have an
isomorphism HdR 2 (Z) −→ H0 dR (Z) given by ∫
[ω] ↦→
Also, HdR n (Z) = 0 for every n ≥ 3. For
cohomology is given by
Z
ω
any compact and connected Riemann surface Z g, the middle
H 1 dR (Z g) = π 1 (Z g )/ 〈commutator subgroup〉 = Z a1 ⊕ Z b1 ⊕ · · · ⊕ Z ag ⊕ Z bg
Theorem 2.9.2. Let ω be a holomorphic differential (←→ 1-form). Then ω is d-closed (hence [ω] ∈ H 1 dR (Z)).
Proof: Locally, ω = f(z)dz where f ∈ O. Then
dω = (∂ + ∂)ω = ∂ω + ∂ω =
( ) ( )
∂f
∂f
∂z dz ∧ dz +
∂z dz ∧ dz
= ∂f ∧ dz + ∂f ∧ d = 0 + 0dz ∧ dz, since f is holomorphic.
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