Kähler Manifold
Introduction
A Kähler manifold is a complex manifold equipped with a Kähler metric, named after the German mathematician Erhard Kähler. This type of manifold is a central object of study in several branches of mathematics, including complex geometry, algebraic geometry, and theoretical physics.
Definition
A Kähler manifold is a quadruple (M, J, g, ω), where M is a smooth manifold, J is an almost complex structure, g is a Riemannian metric, and ω is a symplectic form. These components must satisfy certain compatibility conditions. Specifically, the manifold must be Hermitian (meaning that the metric is compatible with the complex structure), and the associated 2-form must be closed.
Properties
Kähler manifolds have many remarkable properties that make them particularly interesting to mathematicians and physicists. Some of these properties include:
- Closedness of the Kähler form: The Kähler form ω associated with a Kähler metric is closed, meaning that its exterior derivative is zero. This property is crucial for many applications in complex and algebraic geometry.
- Ricci curvature: The Ricci curvature of a Kähler manifold has special properties, which are of fundamental importance in the study of complex manifolds. In particular, the Ricci form is closed, and its cohomology class is an invariant of the complex structure.
- Hodge theory: On a compact Kähler manifold, the Hodge decomposition holds. This is a decomposition of the de Rham cohomology of the manifold into pieces associated with the complex structure. This is a fundamental tool in algebraic geometry and complex manifold theory.
Examples
There are many examples of Kähler manifolds in mathematics. Some of the most important ones include:
- Complex Euclidean space: The simplest example of a Kähler manifold is complex Euclidean space, equipped with its standard metric, complex structure, and symplectic form.
- Projective space: Any complex projective space, equipped with the Fubini-Study metric, is a Kähler manifold. These spaces play a fundamental role in algebraic geometry.
- Toric varieties: Toric varieties are examples of Kähler manifolds that are particularly important in combinatorics and symplectic geometry.
Kähler Manifolds in Physics
Kähler manifolds play a significant role in theoretical physics, particularly in string theory and supersymmetry. The reason for this is that the supersymmetry transformations preserve the Kähler structure of the manifold. This makes Kähler manifolds a natural setting for these theories.