Complex Manifold

A complex manifold is a manifold M whose coordinate charts are open subsets of C^n and the transition functions between charts are holomorphic functions. Naturally, a complex manifold of dimension n also has the structure of a real smooth manifold of dimension 2n.

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

See also

Algebraic Variety, Conformal Mapping, Holomorphic Function, Manifold, Riemann Surface, Stein Manifold

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Complex Manifold." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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