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

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