Complex Structure를 보존하는 Isomorphism

Definition 1. (complex manifold) A complex manifold of dimension $n$ is a topological space $X$ which is locally modeled on open subsets of $\mathbb{C}^n$, with coordinate changes given by holomorphic functions.

Definition 2. (holomorphic function) Let $U \subset \mathbb{C}^n$ be open. A function $f: U \to \mathbb{C}$ is holomorphic if it is complex differentiable in each variable locally. For a map $F: U \to \mathbb{C}^m$, $F$ is holomorphic if each component function is holomorphic.

Definition 3. (isomorphism of elliptic curves) For $E_i = \mathbb{C}/\Lambda_i$, an isomorphism $E_1 \to E_2$ is a bijective holomorphic group homomorphism whose inverse is also holomorphic.

In terms of lattices, $E_1$ and $E_2$ are isomorphic as complex elliptic curves if and only if there exists $\alpha \in \mathbb{C}^{\times}$ such that \(\Lambda_2 = \alpha \Lambda_1.\)