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

TL;DR

The paper addresses a core question about smooth Fano varieties with nef tangent bundles by proving an abundance-type equivalence: nefness of the tangent bundle $T_X$ is equivalent to the tautological line $\mathscr O_{\mathbb P T_X}(1)$ being big and semiample, yielding a weak form of the Campana-Peternell conjecture. The approach blends Viehweg’s fiber product trick, augmented base locus analysis, and an intricate MMP-based uniruledness argument to control base loci and derive positivity for $\mathscr O_{\mathbb P T_X}(1)$. By relating Chern and Segre characteristics and exploiting subadjunction and multiplier-ideal techniques, the work connects nefness with stronger positivity notions, advancing the understanding of CP-manifolds. The results provide a framework for approaching conjectures on the structure and classification of Campana-Peternell manifolds and highlight deep interactions between tangent-bundle positivity, base loci, and MMP methods.

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).

Theorems & Definitions (28)

• Conjecture : Campana-Peternell conjecture
• Theorem A
• Theorem B
• Theorem 1.1: BCHM10
• Theorem 1.2: Bir12
• Proposition 1.3
• proof
• Proposition 1.4
• proof
• Lemma 1.5