Positivity of tangent sheaves of projective varieties -- the structure of MRC fibrations

In this paper, we extend the structure theorem for smooth projective varieties with nef tangent bundle to projective klt varieties whose tangent sheaf is either positively curved or almost nef. Specifically, we show that such a variety $X$, up to a finite quasi-\'etale cover, admits a rationally connected fibration $X \to A$ onto an abelian variety $A$. For the proof, we develop the theory of positivity of coherent sheaves on projective varieties. As applications, we establish some relations between the geometric properties and positivity of tangent sheaves.