A decomposition theorem for Q-Fano Kahler-Einstein varieties

Let X be a Q-Fano variety admitting a Kahler-Einstein metric. We prove that up to a finite quasietale cover, X splits isometrically as a product of Kahler-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 TX by OX is polystable with respect to the anticanonical polarization.