Holomorphic map

Summary

A holomorphic map from an open subset of $\C^n$ to $\C^m$ is a map whose $m$ components are holomorphic functions of several variables. Holomorphic maps are the morphisms of several complex variables and of complex manifolds. Their derivative at each point is a complex linear map, given by the complex Jacobian matrix $(\partial F_k / \partial z_j)$. For $n = m = 1$ they are the holomorphic functions of one variable, and holomorphic maps between Riemann surfaces are the same notion for complex curves.