nLab
complex manifold

Idea

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

• a smooth manifold locally isomorphic to ${ℂ}^n$ whose transition functions are holomorphic functions.

• equivalently: a smooth manifold equipped with an integrable almost complex structure

Properties

Covers

Proposition

Every complex manifold admits a good open cover in ${\mathrm{Disk}}_{\mathrm{cmpl}}$.

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

• Stein manifolds

Related concepts

• complex analytic space

• Dolbeault complex

References

For instance

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

• Zachary Maddock, Dobeault cohomology (pdf)