Semi-stability of the tangent sheaf of singular varieties

TL;DR

This paper proves the polystability of the logarithmic tangent sheaf for certain singular varieties, extending known results to broader classes like stable varieties and Calabi-Yau varieties, with implications for their geometric structure.

Contribution

It generalizes Enoki's theorem to log canonical pairs with ample canonical bundle and applies the result to stable and Calabi-Yau varieties, establishing their tangent sheaf polystability.

Findings

Proves polystability of the logarithmic tangent sheaf for log canonical pairs with ample canonical bundle.

Extends the polystability result to stable varieties and singular Calabi-Yau varieties.

Provides new tools for understanding the geometry of singular varieties through tangent sheaf stability.

Abstract

The main goal of this paper is to prove the polystability of the logarithmic tangent sheaf $\mathscr T_X(-D)$ of a log canonical pair $(X,D)$ whose canonical bundle $K_X+D$ is ample, generalizing in a significant way a theorem of Enoki. We apply this result and the techniques involved in its proof to get a version of this theorem for stable varieties (the higher dimensional analogue of Deligne-Mumford's stable curves) and to prove the polystability with respect to any polarization of the tangent sheaf of a singular Calabi-Yau variety.