Fano varieties with split tangent sheaf

TL;DR

This paper studies mildly singular Fano varieties with tangent sheaves that split into direct sums, showing that these factors are integrable and lead to a product structure on a cover of the variety.

Contribution

It proves that the direct factors of the tangent sheaf in such Fano varieties are algebraically integrable, inducing a product structure on a quasi-étale cover.

Findings

Direct factors are algebraically integrable

Induces a product structure on a cover

Applicable to mildly singular Fano varieties

Abstract

Let $X$ be a mildly singular Fano variety such that the tangent sheaf is a direct sum. We show that the direct factors are algebraically integrable, so the infinitesimal decomposition induces a product structure on a quasi-\'etale cover of $X$.