Canal-U

C. Araujo - Foliations and birational geometry (Part 4)

In the last few decades, much progress has been made in birational algebraic geometry. The general viewpoint is that complex projective manifolds should be classified according to the behavior of their canonical class. As a result of the minimal model program (MMP), every complex projective manifold can be built up from 3 classes of (possibly singular) projective varieties, namely,  varieties $X$ for which  $K_X$ satisfies $K_X0$. Projective manifolds $X$ whose anti-canonical class $-K_X$ is ample are called Fano manifolds.

Techniques from the MMP have been successfully applied to the study of global properties of holomorphic foliations. This led, for instance, to Brunella's birational classification of foliations on surfaces, in which the canonical class of the foliation plays a key role. In recent years, much progress has been made in higher dimensions. In particular, there is a well developed theory of Fano foliations, i.e., holomorphic foliations $F$ on complex projective varieties with ample anti-canonical class  $-K_F$.

The first half of this mini-course will be devoted to explaining the MMP and reviewing some aspects of the theory of Fano manifolds. In the second half, we will explore the foliated counterpart of this theory, with a special emphasis on Fano foliations.