nLab
complex manifold

Context

Complex geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Manifolds and cobordisms

Contents

Idea

A complex manifold is a manifold holomorphically modeled on polydiscs DD in n\mathbb{C}^n (complexified nn-dimensional cartesian space):

Properties

Covers

Proposition

Every complex manifold admits a good open cover in Disk cmplDisk_{cmpl}.

For instance (Maddock, lemma 3.2.8).

Examples

Complex 1-dimensional: Riemann surfaces

A complex manifold of complex dimension 1 is called a Riemann surface.

Calabi-Yau manifolds

A complex manifold whose canonical bundle is trivializable is a Calabi-Yau manifold. In complex dimension 2 this is a K3 surface.

Other examples

References

Textbook accounts include

Lectures notes inclide

  • Stefan Vandoren, Lectures on Riemannian Geometry, Part II: Complex Manifolds (pdf)

  • Zachary Maddock, Dobeault cohomology (pdf)

Revised on July 17, 2014 14:19:58 by Todd Trimble (67.81.95.215)