Space of Holomorphic Differentials A smooth (1, 0) form ω on a Riemann surface X is locally of the form ω = f (z, z)dz

1. Space of Holomorphic Differentials A smooth (1, 0) form ω on a Riemann surface X is locally of the form

ω = f (z, z)dz.

We say that ω is holomorphic if ∂ω = 0, where

∂ω = ∂f

∂zdz ∧ dz.

In other words, ∂f /∂z = 0. We write f (z, z) by f (z) when ∂f /∂z = 0. Hence a holomorphic differential ω is locally represented by f (z)dz where f (z) is holomorphic. The space of holomorphic differentials on X is denoted by H⁰(X, ω_X).

Theorem 1.1. On a compact Riemann surface of genus g, the complex vector space H⁰(X, ωX) has dimension g.

Proof. Denote H = H⁰(X, ωX) and let H be the space of complex valued harmonic one- forms on X. Then

H = H ⊕ H.

Here H is the space of anti-holomorphic differentials¹ on X. The map H → H defined by ω 7→ ω is an R-linear isomorphism. Hence dimRH = dim_RH. Now,

dim_CH = 1

2dim_RH = 1

4dim_RH = 1

2dim_CH.

Hence we only need to compute the complex dimension of H. By Hodge theorem, H ∼= H_dR¹ (X, C).

By Poincare duality, we know

H_dR¹ (X, C) ∼= H1(X, C).

Since H1(X, C) = H1(X, Z) ⊗ZC, H1(X, Z) is a 2g-dimensional complex vector space. Thus

dim_CH = 2g. This implies that dim_CH = g. 

1A smooth (0, 1)-form is called anti-holomorphic if ∂ω = 0.

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