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. On a Kähler manifold X there exists Kähler potential and the Levi-Civita connection corresponding to the metric of X gives rise to a connection on the canonical line bundle.

Smooth projective algebraic varieties are examples of Kähler manifolds. By Kodaira embedding theorem, Kähler manifolds that have a positive line bundle can always be embedded into projective spaces.

They are named after German mathematician Erich Kähler.

Definitions

Since Kähler manifolds are naturally equipped with several compatible structures, there are many equivalent ways of creating Kähler forms.

Symplectic viewpoint

A Kähler manifold is a symplectic manifold equipped with an integrable almost-complex structure which is compatible with the symplectic form.

Complex viewpoint

A Kähler manifold is a Hermitian manifold whose associated Hermitian form is closed. The closed Hermitian form is called the Kähler metric.