Hermitian manifold

Summary

A Hermitian manifold is a complex manifold with a Riemannian metric that is compatible with multiplication by $i$ on tangent vectors. It is the complex analogue of a Riemannian manifold, and every complex manifold admits such a metric. Each Hermitian metric has a fundamental $2$-form, and when this form is closed the manifold is a Kähler manifold. Kähler manifolds are the setting of Hodge theory and of much of complex algebraic geometry.