Editing Complex geometry (section)

=== Calabi–Yau manifolds ===
{{Main article|Calabi–Yau manifold}}

[[File:CalabiYau5.jpg|thumb|A real two-dimensional slice of a quintic [[Calabi–Yau manifold|Calabi–Yau]] threefold]]
As mentioned, a particular class of Kähler manifolds is given by Calabi–Yau manifolds. These are given by Kähler manifolds with trivial canonical bundle <math>K_X = \Lambda^n T_{1,0}^* X</math>. Typically the definition of a Calabi–Yau manifold also requires <math>X</math> to be compact. In this case [[Shing-Tung Yau|Yau's]] proof of the [[Calabi conjecture]] implies that <math>X</math> admits a Kähler metric with vanishing [[Ricci curvature]], and this may be taken as an equivalent definition of Calabi–Yau. 

Calabi–Yau manifolds have found use in [[string theory]] and [[Mirror symmetry (string theory)|mirror symmetry]], where they are used to model the extra 6 dimensions of spacetime in 10-dimensional models of string theory. Examples of Calabi–Yau manifolds are given by [[elliptic curve]]s, K3 surfaces, and complex [[Abelian varieties]].