COMPLEX GEOMETRY Course notes

2.14 Review
Theorem 2.14.1 (Orthogonal decomposition). For every ω ∈ A 1 = space of differntial 1-forms, there exists
a unique decomposition
ω = ω h + df + ∗dg
where ω h is a bounded harmonic, f, g ∈ A 0 = space of smooth functions.
We denote A = the space of C ∞ -functions.
Proof: Let H be the space of harmonic differentials. Then H is orthogonal to both E = dA 0 and
∗E = ∗dA 0 . This fact follows easily using the L 2 -inner product and the equality ∗∗ = (−1) k , for
Riemann surfaces one has (−1) k = 1. We have
H⊥E⊥ ∗ E ⊆ A 1 .
Taking completion, we have
Ȟ⊥○Ě ⊕ ∗Ě = A1
where Ȟ⊥○Ě is the orthogonal (Ȟ ⊥ Ě) direct sum of Ȟ and Ě. By the Weyl’s Lemma, we have H = Ȟ.
Lemma 2.14.1. Any distribution (1-form) T (1-current) with ∆T = 0 is the distribution of some differential
function f, i.e., T = T f where
∫
T f [h] = hf
and h is compactly supported in U ⊂⊂ Z, where Z is a Riemann surface.
U
Corollary 2.14.1 (Hodge decomposition). HdR 1 (Z) = Ω ⊕ Ω = H.
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