Twisted cotangent sheaves and a Kobayashi-Ochiai theorem for foliations

TL;DR

This paper proves that the twisted cotangent sheaf of a normal projective variety is generically nef unless the variety is projective space, and establishes a Kobayashi-Ochiai type inequality for foliations.

Contribution

It introduces a new nefness property of twisted cotangent sheaves and extends the Kobayashi-Ochiai theorem to foliations on projective varieties.

Findings

Twisted cotangent sheaf is generically nef unless X is projective space.

A Kobayashi-Ochiai type inequality for foliations is established.

Provides bounds on the index of foliations in terms of their rank.

Abstract

Let X be a normal projective variety, and let A be an ample Cartier divisor on X. We prove that the twisted cotangent sheaf $\Omega_X \otimes A$ is generically nef with respect to the polarisation A unless X is a projective space. As an application we prove a Kobayashi-Ochiai theorem for foliations: if $F \subsetneq T_X$ is a foliation of rank r such that $det F \equiv i_F A$, then we have $i_F \leq r$.