A complex manifold is a manifold with an atlas of charts into whose transition maps are holomorphic, equivalently carrying an integrable complex structure with . A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated Kähler form is closed, .
The closure condition binds a trinity into mutual compatibility: a Riemannian metric , a complex structure , and a symplectic form . Any two of these determine the third through the relation , so a Kähler structure is simultaneously a piece of Riemannian, complex, and symplectic geometry. See riemannian-geometry for the metric side.
The Ricci form is a closed real -form representing in de Rham cohomology, so a Kähler manifold is Ricci-flat exactly when . The Kähler class is the cohomology class , an invariant of the metric within its cohomological family.
Prototypes are with its flat metric, complex projective space with the Fubini-Study metric, and, by restriction, every smooth projective variety.