Editing Complex manifold (section)

== Kähler and Calabi–Yau manifolds ==
One can define an analogue of a [[Riemannian metric]] for complex manifolds, called a [[Hermitian metric]]. Like a Riemannian metric, a Hermitian metric consists of a smoothly varying, positive definite inner product on the tangent bundle, which is Hermitian with respect to the complex structure on the tangent space at each point. As in the Riemannian case, such metrics always exist in abundance on any complex manifold. If the skew symmetric part of such a metric is [[Symplectic geometry|symplectic]], i.e. closed and nondegenerate, then the metric is called [[Kähler manifold|Kähler]]. Kähler structures are much more difficult to come by and are much more rigid. 

Examples of [[Kähler manifold]]s include smooth [[projective varieties]] and more generally any complex submanifold of a Kähler manifold. The [[Hopf manifold]]s are examples of complex manifolds that are not Kähler. To construct one, take a complex vector space minus the origin and consider the action of the group of integers on this space by multiplication by exp(''n''). The quotient is a complex manifold whose first [[Betti number]] is one, so by the [[Hodge theory]], it cannot be Kähler.

A [[Calabi–Yau manifold]] can be defined as a compact [[Ricci-flat manifold|Ricci-flat]] Kähler manifold or equivalently one whose first [[Chern class]] vanishes.