Abstract: I will discuss recent work with D.Greb and
S. Kebekus on singular varieties which are in some
sense Ricci flat. Those normal projective varieties
with a certain type of singularities occur naturally as end-products (minimal models) in the Mori theory. In particular I will discuss a decomposition of the tangent bundle into stable subsheaves and perspectives for the decomposition of the variety itselfin the spirit of Beauville-Bogomolov-de Rham.