Holomorphic map

A map $f:U\to\mathbb C^m$, where $U\subseteq\mathbb C^n$ is open, is holomorphic if it is complex differentiable in all variables; equivalently, its component functions are locally given by convergent complex power series.

For complex manifolds, a map is holomorphic when its expression in every pair of complex coordinate charts is holomorphic in this Euclidean sense. A bijective holomorphic map with holomorphic inverse is a biholomorphism.