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I am studying algebraic curves through the book "Intr. to Algebraic Curves - Griffiths". The following definition puzzled too:


Definition: Suppose $C$ is a Riemann surface. Then a holomorphic differential $\omega$ is by definition a family ${(U_i, z_i, \omega_i)}$ such that

1)${(U_i,z_i)}$ is a holomorphic covering of $C$ and $\omega=f_i(z_i)dz_i$ where $f_i$ is holomorphic function on $U_i$,

2)if $z_i=\phi_{ij}(z_j)$ is a coordinate transformation on $U_i\cap U_j$, then

$f_i(\phi_{ij}(z_j))\frac{d\phi_{ij}(z_j)}{dz_j}=f_j(z_j)$


Question: Since domain of $f_i$ and $z_i$ are $U_i$; so how can we talk about $f_i(z_i)$?

Can one help to change the definition slightly?

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1 Answer 1

That seems like a somewhat confused definition. Here's one which is hopefully clearer.

Definition. Let $C$ be a Riemann surface. A holomorphic differential $\omega$ is given by a family $(U_i, z_i, \omega_i)$ such that:

  1. The $U_i$ together form an open cover of $C$, and
  2. each $z_i : U_i \to V_i$ is a homeomorphism onto $V_i$, and
  3. on each intersection $U_i \cap U_j$, the transition map $\phi_{ij} : z_j[U_i \cap U_j] \to z_i[U_i \cap U_j]$ is holomorphic, and
  4. each $\omega_i : V_i \to \mathbb{C}$ is holomorphic, and
  5. $(\omega_i \circ \phi_{ij}) \cdot \phi'_{ij} = \omega_j$.

Informally, $\omega$ is given in coordinates by $\omega_i$ and the $\omega_i$ must satisfy a compatibility condition similar to the chain rule. Let $f : C \to \mathbb{C}$ be a holomorphic function, given in coordinates by $f_i : V_i \to \mathbb{C}$. Then, we can form the holomorphic differential $\mathrm{d}f$ by taking $\omega_i = f'_i$; indeed, the $f_i$ satisfy the obvious compatibility condition $$f_i \circ \phi_{ij} = f_j$$ so differentiating both sides, we obtain $$(f'_i \circ \phi_{ij}) \cdot \phi'_{ij} = f'_j$$ which is precisely the compatibility condition we need for $\omega_i$ to be the components of a holomorphic differential.

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