Mathematics

Given a smooth complex projective variety X, many geometrical properties of X depend on its canonical line bundle, the determinant of the cotangent bundle of X. Varieties with positive/trivial/negative canonical line bundle play a distinguished role in the birational classification of complex projective varieties, in the framework of the Minimal Model Program. In particular, X is a Fano variety if its canonical line bundle is negative (more precisely, its dual is ample); this yields many nice properties and a rich geometry. We will review the role of Fano varieties in the framework of the Minimal Model Program, and then explain some of their properties.