complex analytic manifold


A manifold M is called a complex analytic manifold (or sometimes just a complex manifold) if the transition functions are holomorphic.


A subset NM is called a complex analytic submanifold of M if N is closed in M and if for every point zN there is a coordinate neighbourhood U in M with coordinates z1,,zn such that UN={pUzd+1(p)==zn(p)} for some integer dn.

Obviously N is now also a complex analytic manifold itself.

For a complex analytic manifold, dimension always means the complex dimension, not the real dimension. That is M is of dimension n when there are neighbourhoods of every point homeomorphic to n. Such a manifold is of real dimension 2n if we identify n with 2n. Of course the tangent bundleMathworldPlanetmath is now also a complex vector space.

A function f is said to be holomorphic on M if it is a holomorphic function when considered as a function of the local coordinates.

Examples of complex analytic manifolds are for example the Stein manifoldsMathworldPlanetmath or the Riemann surfaces. Of course also any open set in n is also a complex analytic manifold. Another example may be the set of regular pointsMathworldPlanetmath of an analytic set.

Complex analytic manifolds can also be considered as a special case of CR manifolds where the CR dimension is maximal.

Complex manifolds are sometimes described as manifolds carrying an or . This refers to the atlas and transition functions defined on the manifold.


  • 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
  • 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title complex analytic manifold
Canonical name ComplexAnalyticManifold
Date of creation 2013-03-22 15:04:40
Last modified on 2013-03-22 15:04:40
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 6
Author jirka (4157)
Entry type Definition
Classification msc 32Q99
Synonym complex manifold
Defines complex analytic submanifold
Defines complex submanifold
Defines analytic structure
Defines holomorphic structure