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

TL;DR

This paper extends the structure theorem for smooth projective varieties with nef tangent bundles to certain singular varieties, showing they admit a rationally connected fibration over an abelian variety, and develops positivity theory for coherent sheaves.

Contribution

It generalizes the structure theorem to projective klt varieties with positively curved or almost nef tangent sheaves, introducing new positivity techniques.

Findings

Varieties with positive or almost nef tangent sheaves admit a rationally connected fibration over an abelian variety.

Develops a theory of positivity for coherent sheaves on projective varieties.

Establishes relations between geometric properties and tangent sheaf positivity.

Abstract

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.