nLab
complex manifold

Context

Complex geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

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)