Complex algebraic geometry

Complex Fano varieties

A complex Fano variety is a complex algebraic variety with ample anti-canonical line bundle (that is, ${\displaystyle K_{X}^{*}}$ 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 ${\displaystyle \mathbb {CP} ^{n}}$ where ${\displaystyle K={\mathcal {O}}(-n-1)}$, and smooth hypersurfaces of ${\displaystyle \mathbb {CP} ^{n}}$ of degree less than ${\displaystyle n+1}$.