Complex Manifold Space
(Redirected from Complex Manifold)
Jump to navigation
Jump to search
A Complex Manifold Space is a manifold space that locally resembles complex Euclidean space and allows the application of complex analytical methods.
- Context:
- It can (typically) have each point with a neighborhood that is biholomorphically equivalent to an open subset of the complex coordinate space \( \mathbb{C}^n \).
- It can (typically) be covered by a set of coordinate charts with holomorphic transition functions.
- It can (often) facilitate the study of holomorphic functions and their generalizations.
- ...
- ...
- It can serve as the setting for advanced topics in complex geometry, differential geometry, and algebraic geometry.
- It can be compact, such as a Complex Projective Variety, or non-compact like the Complex Plane.
- ...
- Example(s):
- Complex Planes.
- Riemann Surface of complex dimension 1
- higher-dimensional complex manifold spaces, such as:mplex manifold structure.
- a Complex Torus, which is a higher-dimensional analog of an elliptic curve.
- a Complex Projective Space [math]\displaystyle{ (\mathbb{CP}^n) }[/math], which is important in projective geometry and complex geometry.
- Counter-Example(s):
- Real Manifolds, which lack the complex structure required for complex manifolds.
- Affine Varietys that are not equipped with the same complex analytical properties necessary for complex manifolds.
- See: Holomorphic Function, Riemann Surface, Complex Analysis, Differential Geometry, Algebraic Geometry.