A decomposition theorem for $\mathbb{Q}$-Fano Kähler-Einstein varieties

Abstract

Let $X$ be a $\mathbb{Q}$-Fano variety admitting a Kähler-Einstein metric. We prove that up to a finite quasi-étale cover, $X$ splits isometrically as a product of Kähler-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 ${\mathcal{O}}_X$ is semistable with respect to the anticanonical polarization.

arXiv comment27 pages

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Cited by in corpus (1)

• Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor