Complex Manifold

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.