# nLab complex manifold

complex geometry

### Examples

#### Differential geometry

differential geometry

synthetic differential geometry

## Applications

#### Manifolds and cobordisms

manifolds and cobordisms

# Contents

## Idea

A complex manifold is a manifold modeled on $\mathbb{C}^n$ (the complex $n$-dimensional complex line):

## Properties

### Covers

###### Proposition

Every complex manifold admits a good open cover in $Disk_{cmpl}$.

## 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.

## References

For instance

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

• Zachary Maddock, Dobeault cohomology (pdf)

Revised on July 17, 2013 19:53:28 by Urs Schreiber (82.169.65.155)