Let $M$ be a complex manifold with integrable complex structure $J$

Suppose $M$ is also a Riemannian manifold with metric tensor $g$ such that $\forall_{X,Y} g(X,Y) = g(JX,JY)$ We say that $g$ is an Hermitian metric tensor.

A differentiable manifold $M$ is said to be a Kähler manifold iff all the following conditions are verified:

• $M$ is a complex manifold with complex structure $J$
• $M$ is a Riemannian manifold with an Hermitian metric $g$
• $J$ is covariantly constant with regard to the Levi-Civita connection ($\nabla J = 0$

Kähler manifolds are symplectic in a natural way with symplectic form defined by $\omega(X,Y) = g(JX,Y)$