Complex manifold is a manifold with holomorphic transition functions. For a complex manifold dim(M)=m then the manifold is homeomorphic to or .

Equivalence:

• Real manifolds - diffeomorphism
• complex manifolds - biholomorphic

Complex structures are the number of ways to biholomorphically connect coordinate patches.

Examples:

• , . The map to the plane is: for , the map to the plane is: . This gives . This is analtyic everywhere except the poles as required. This is the unique complex structure
• , the torus

source psfile jl@crush.caltech.edu index holomorphic_principle_bundle holomorphic_vector_bundle almost_complex_structure character_class_example canonical_line_bundle splitting_principle exterior_derivative complex_laplacian hermitian_metric complex_metric hodge_number