Kähler manifold

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was introduced by Erich Kähler in 1933.

Every smooth complex projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics.

Since Kähler manifolds are equipped with several compatible structures, they can be described from different points of view:

A Kähler manifold is a symplectic manifold (X,ω) equipped with an integrable almost-complex structure J which is compatible with the symplectic form ω, meaning that the bilinear form

on the tangent space of X at each point is symmetric and positive definite (and hence a Riemannian metric on X).

A Kähler manifold is a complex manifold X with a Hermitian metric h whose associated 2-form ω is closed. In more detail, h gives a positive definite Hermitian form on the tangent space TX at each point of X, and the 2-form ω is defined by