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