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

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 ${\mathcal{O}}_{\mathbb{P}{T}_X}(1)$ is so, by which we establish a weak form of the Campana-Peternell conjecture (Camapan-Peternell, 1991).

Version 2: this version adds a prominent notice on the first page regarding a gap found in the original manuscript (V1) and an erratum at the end to explain this error in details. The main text is otherwise unchanged. A corrected new version (V3) will be uploaded within approximately 1 month