Stability properties of adapted tangent sheaves on K\"ahler--Einstein log Fano pairs

TL;DR

This paper proves the polystability of adapted tangent sheaves and canonical extensions on log Fano pairs with K"ahler--Einstein metrics, advancing understanding of their geometric stability properties.

Contribution

It establishes the polystability of adapted tangent sheaves and canonical extensions on log Fano pairs with singular K"ahler--Einstein metrics, a novel result in complex geometry.

Findings

Adapted tangent sheaves are polystable under certain conditions.

Adapted canonical extensions are polystable with respect to specific classes.

Results apply to log Fano pairs with singular K"ahler--Einstein metrics.

Abstract

Let $(X, \Delta)$ be a log Fano pair with standard coefficients endowed with a singular K\"ahler--Einstein metric. We show that the adapted tangent sheaf $\mathcal{T}_{X, \Delta, f}$ and the adapted canonical extension $\mathcal{E}_{X, \Delta, f}$ are polystable with respect to $f^*c_1(X, \Delta)$ for any strictly $\Delta$-adapted morphism $f: Y \to X$.