Kähler manifold

Summary

A Kähler manifold is a complex manifold with a Hermitian metric whose associated $2$-form is closed. Its Riemannian, complex and symplectic structures are then compatible in the strongest way. Complex projective space and all its complex submanifolds are Kähler, so Kähler geometry contains smooth projective algebraic geometry. Compact Kähler manifolds satisfy strong topological restrictions, such as the Hodge decomposition of their cohomology and the evenness of odd Betti numbers.