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.
