Complex manifold

Introduction

A complex manifold is a type of manifold equipped with an atlas of charts to the open unit disk in C^n, such that the transition maps are holomorphic. It is a fundamental concept in the field of complex geometry, which extends and refines the ideas of differential geometry in the context of complex vector spaces.

Definition

Formally, a complex manifold is a manifold of even dimension, together with a maximal atlas whose transition functions are holomorphic. More specifically, if M is a complex manifold of complex dimension n, then it is a manifold of real dimension 2n, and the charts that make up its atlas map into C^n, the n-dimensional complex space. The transition functions, which are the functions that translate between different charts, are required to be holomorphic.

Structure

The structure of a complex manifold is determined by its holomorphic functions. These functions form a sheaf of rings over the manifold, known as the structure sheaf. The local rings of the structure sheaf at each point in the manifold are isomorphic to the ring of germs of holomorphic functions at that point. This gives the manifold a rich algebraic structure, which is the basis for much of the study of complex manifolds.

Types of Complex Manifolds

There are several different types of complex manifolds, distinguished by their additional structure or properties.

Kähler Manifolds

A Kähler manifold is a complex manifold that has a Hermitian metric whose associated two-form is closed. This condition, known as the Kähler condition, implies that the manifold has a number of nice geometric properties. For example, Kähler manifolds have vanishing first Chern class, which means that they are topologically trivial in a certain sense.

Complex Tori

A complex torus is a complex manifold that is diffeomorphic to a real torus of twice the dimension. These manifolds are important in the study of algebraic geometry, as they provide examples of non-algebraic complex manifolds.

Calabi-Yau Manifolds

Calabi-Yau manifolds are Kähler manifolds that have a trivial canonical bundle. These manifolds are of particular interest in string theory, as they are the spaces on which strings are conjectured to propagate.

Properties

Complex manifolds have a number of distinctive properties that set them apart from other types of manifolds.

Holomorphic Functions

The holomorphic functions on a complex manifold are much more rigid than the smooth functions on a real manifold. For example, the identity theorem for holomorphic functions implies that a holomorphic function on a connected complex manifold is determined by its values on any arbitrarily small open set.

Complex Structure

The complex structure of a complex manifold gives it a natural orientation. This is in contrast to real manifolds, which need not be orientable.

Chern Classes

The Chern classes of a complex manifold are topological invariants that measure the extent to which the tangent bundle of the manifold deviates from being trivial. These classes play a central role in the study of complex manifolds, particularly in the theory of characteristic classes.

Applications

Complex manifolds have applications in several areas of mathematics and theoretical physics.

Algebraic Geometry

In algebraic geometry, complex manifolds serve as a natural setting for the study of algebraic varieties. The field of complex algebraic geometry is largely concerned with the study of complex manifolds that are also algebraic varieties.

Differential Geometry

In differential geometry, complex manifolds provide a rich source of examples and counterexamples. They also serve as a testing ground for conjectures and theories.

Theoretical Physics

In theoretical physics, complex manifolds play a central role in string theory. In particular, the extra dimensions in which strings propagate are often modeled as compact complex manifolds, particularly Calabi-Yau manifolds.