A complex manifold is a manifold M whose coordinate charts are open subsets of C^n and the transition functions between charts are holomorphic functions. Naturally, a complex manifold of dimension n also has the structure of a real smooth manifold of dimension 2n. A function f:M->C is holomorphic if it is holomorphic in every coordinate chart. Similarly, a map f:M->N is holomorphic if its restrictions to coordinate charts on N are holomorphic. Two complex manifolds M and N are considered equivalent if there is a map f:M->N which is a diffeomorphism and whose inverse is holomorphic.

algebraic variety | conformal mapping | holomorphic function | manifold | Riemann surface