An abundance-type result for the tangent bundles of smooth Fano varieties

TL;DR

This paper proves a characterization of the tangent bundle of smooth Fano varieties, showing it is nef if and only if it is big and semiample, thus providing a weak form of the Campana-Peternell conjecture.

Contribution

It establishes a new equivalence condition for the tangent bundle of smooth Fano varieties, linking nefness with bigness and semi-ampleness.

Findings

Tangent bundle $T_X$ is nef iff it is big and semiample for smooth Fano varieties.

Establishes a weak form of the Campana-Peternell conjecture.

Provides new geometric criteria for tangent bundle positivity.

Abstract

In this paper we prove the following abundance-type result: for any smooth Fano variety $X$, the tangent bundle $T_X$ is nef if and only if it is big and semiample in the sense that the tautological line bundle $\mathscr{O}_{\mathbb{P}T_X}(1)$ is so, by which we establish a weak form of the Campana-Peternell conjecture (Camapan-Peternell, 1991).