# On generic nefness of tangent sheaves

## TL;DR

This paper proves that the tangent bundle of certain projective manifolds with nef anticanonical class is generically nef, confirming a conjecture, and explores implications for the second Chern class and related positivity conditions.

## Contribution

It establishes the generic nefness of tangent bundles for manifolds with nef anticanonical class, confirming Peternell's conjecture and analyzing positivity conditions.

## Key findings

- Tangent bundle is generically nef for manifolds with nef anticanonical class.
- Second Chern class has non-negative intersections with ample divisors.
- Conditions for strict positivity are identified and related to Yau's question.

## Abstract

We show that the tangent bundle of a projective manifold with nef anticanonical class is generically nef. That is, its restriction to a curve cut out by general sufficiently ample divisors is a nef vector bundle. This confirms a conjecture of Peternell. As a consequence, the second Chern class of such a manifold has non-negative intersections with ample divisors. We also investigate under which conditions these positivities are strict, and answer a question of Yau.