A decomposition theorem for $\mathbb Q$-Fano K\"ahler-Einstein varieties

TL;DR

This paper proves that $Q$-Fano K"ahler-Einstein varieties decompose into products of stable KE varieties after a finite cover, using a general splitting theorem for algebraically integrable foliations.

Contribution

It establishes a decomposition theorem for $Q$-Fano K"ahler-Einstein varieties, revealing their structure via a splitting after finite quasi-étale covers.

Findings

Varieties split as products of KE $Q$-Fano varieties after finite cover

Tangent sheaf stability with respect to the anticanonical polarization

Canonical extension of $T_X$ is semistable

Abstract

Let $X$ be a $\mathbb Q$-Fano variety admitting a K\"ahler-Einstein metric. We prove that up to a finite quasi-\'etale cover, $X$ splits isometrically as a product of K\"ahler-Einstein $\mathbb Q$-Fano varieties whose tangent sheaf is stable with respect to the anticanonical polarization. This relies among other things on a very general splitting theorem for algebraically integrable foliations. We also prove that the canonical extension of $T_X$ by $\mathscr O_X$ is semistable with respect to the anticanonical polarization.