Stability properties of adapted tangent sheaves on Kähler--Einstein log Fano pairs

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