Editing Complex geometry (section)

=== Complex Fano varieties ===
{{Main article|Fano variety}}
A complex [[Fano variety]] is a complex algebraic variety with [[ample line bundle|ample]] anti-canonical line bundle (that is, <math>K_X^*</math> is ample). Fano varieties are of considerable interest in complex algebraic geometry, and in particular [[birational geometry]], where they often arise in the [[minimal model program]]. Fundamental examples of Fano varieties are given by projective space <math>\mathbb{CP}^n</math> where <math>K=\mathcal{O}(-n-1)</math>, and smooth hypersurfaces of <math>\mathbb{CP}^n</math> of degree less than <math>n+1</math>.