nLab complex manifold

Idea

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

• a smooth manifold locally isomorphic to $D \hookrightarrow \mathbb{C}^n$ whose transition functions are holomorphic functions;

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

• equivalently a smooth complex analytic space.

Properties

Covers

Proposition

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

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

• Riemann sphere

• Stein manifolds

Related concepts

• complex analytic space

• domain of holomorphy, Stein manifold, Oka manifold

• Dolbeault cohomology, Bott-Chern cohomology

• holomorphic symplectic manifold

• complex Lie group

• complex orbifold

• complex analytic stack

• complex analytic ∞-groupoid

• hypercomplex manifold

• smooth manifold, topological manifold

References

Textbook accounts:

• Claire Voisin, section 2 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

• Franc Forstnerič, Section 1.5 in: Stein manifolds and holomorphic mappings – The homotopy principle in complex analysis, Springer 2011 (doi:10.1007/978-3-642-22250-4)

  (in the context of the Oka principle)

See also:

• Wikipedia, Complex manifold

With an eye towards application in mathematical physics:

• Mikio Nakahara, Chapter 8 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

Lectures notes:

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

• Zachary Maddock, Dobeault cohomology (pdf)