Editing Complex geometry (section)

=== Kähler manifolds ===
{{Main article|Kähler manifold}}

Complex manifolds may be studied from the perspective of differential geometry, whereby they are equipped with extra geometric structures such as a [[Riemannian metric]] or [[symplectic form]]. In order for this extra structure to be relevant to complex geometry, one should ask for it to be compatible with the complex structure in a suitable sense. A [[Kähler manifold]] is a complex manifold with a Riemannian metric and symplectic structure compatible with the complex structure. Every complex submanifold of a Kähler manifold is Kähler, and so in particular every non-singular affine or projective complex variety is Kähler, after restricting the standard Hermitian metric on <math>\mathbb{C}^n</math> or the [[Fubini-Study metric]] on <math>\mathbb{CP}^n</math> respectively.

Other important examples of Kähler manifolds include [[Riemann surface]]s, [[K3 surface]]s, and [[Calabi–Yau manifold]]s.