A decomposition theorem for ℚ-Fano Kähler–Einstein varieties

Let X be a 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 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 𝒪 X is polystable with respect to the anticanonical polarization.