In this chapter, we consider an additional structure on complex manifolds: a Kähler metric. A complex manifold X can always be equipped with a Hermitian metric, i.e. with a collection of Hermitian metrics, one on each tangent space TX,x varying differentiably with x; the tangent space for each point x is equipped here with the complex structure Jx induced by the complex structure of X. We show this by using partitions of unity and local trivialisations of the tangent bundle as a complex vector bundle.
A Hermitian structure h is a sesquilinear form written
where g is a Riemannian metric and ω is a 2-form called a Kähler form, ...