On Fano foliations

TL;DR

This paper studies Fano foliations on complex projective varieties, especially del Pezzo foliations, proving their algebraic integrability and classifying those with mild singularities, advancing understanding of their structure.

Contribution

It demonstrates the algebraic integrability of del Pezzo foliations and provides a classification for those with mild singularities, highlighting a key exception involving projective space.

Findings

Del Pezzo foliations are algebraically integrable.

Exceptional case: projective space as ambient space.

Classification of del Pezzo foliations with mild singularities.

Abstract

In this paper we address Fano foliations on complex projective varieties. These are foliations whose anti-canonical class is ample. We focus our attention on a special class of Fano foliations, namely del Pezzo foliations on complex projective manifolds. We show that these foliations are algebraically integrable, with one exceptional case when the ambient space is a projective space. We also provide a classification of del Pezzo foliations with mild singularities.