A complex manifold is a manifold whose coordinate charts are open subsets of and the transition functions between charts are holomorphic functions. Naturally, a complex manifold of dimension also has the structure of a real smooth manifold of dimension .

A function is holomorphic if it is holomorphic in every coordinate chart. Similarly, a map is holomorphic if its restrictions to coordinate charts on are holomorphic. Two complex manifolds and are considered equivalent if there is a map which is a diffeomorphism and whose inverse is holomorphic.