Kähler manifolds

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 first studied by the first time by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.

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