COMPLEX GEOMETRY Course notes

Recall that Ω denotes the space of holomorphic 1-forms on S. Since
H 1 dR
∼ =
−→ H 1 (S; Z) ⊗ C = Hom C (H 1 , C)
if S is compact, we have an inclusion Ω ↩→ HdR 1 if S is compact. We have
ω exact =⇒ ω = df =⇒ f holomorphic =⇒ f constant =⇒ ω = 0
Definition 2.10.3. P g = dim(Ω) (geometric genus).
It is known that P g ≥ g. When the equality holds, it is because of the Hodge Decomposition Theorem.
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