Fano manifolds with nef tangent bundles are weakly almost K{\"a}hler-Einstein

TL;DR

This paper proves that Fano manifolds with nef tangent bundles admit a weak form of almost Kähler-Einstein metrics, using regularization techniques and exploring related stability and Chern inequality issues.

Contribution

It establishes the existence of almost Kähler-Einstein metrics on Fano manifolds with nef tangent bundles, a novel result linking positivity conditions to metric properties.

Findings

Existence of almost Kähler-Einstein metrics on these manifolds

Application of regularization theorem for positive (1,1)-currents

Discussion of semistability and Chern inequalities

Abstract

The goal of this short note is to point out that every Fano manifold with a nef tangent bundle possesses an almost K{\"a}hler-Einstein metric, in a weak sense. The technique relies on a regularization theorem for closed positive (1, 1)-currents. We also discuss related semistability questions and Chern inequalities.