On Fano foliations (Q387879)

The paper under review studies foliations with ample anti-canonical bundles and high index (see below for definition) on a smooth projective variety.NEWLINENEWLINEA foliation \(\mathcal{F} \subset T_X\) on a smooth projective variety \(X\) is called a Fano foliation if the anti-canonical class of the foliation \(-K_{\mathcal{F}}=c_1(\mathcal{F})\) is ample. Define the index \(i\) of the foliation to be the largest integer dividing \(-K_{\mathcal{F}}\) in \(\text{Pic}(X)\). Finally the foliation is called a \textit{del Pezzo foliation} if the index \(i\) equals to \(\dim X-2\), a terminology motivated by the classification of Fano manifolds.NEWLINENEWLINEThe paper first shows that del Pezzo foliations are algebraically integrable except in one case, where the ambient space is the projective space (Theorem 1.1).NEWLINENEWLINEThe authors then introduced the notion of singularities of the foliation along a general leaf, mimicking definitions in the minimal model program. Then they show that for a log canonical foliation, either the ambient space has Picard number one, or it is a projective space bundle over a projective space (Theorem 1.3).NEWLINENEWLINEFinally the authors give a classification of del Pezzo foliations on projective space bundles over a projective space (Theorem 1.4).