Stability of Equivariant Logarithmic Tangent Sheaves on Toric Varieties of Picard Rank Two

TL;DR

This paper investigates the conditions under which the logarithmic tangent sheaf on certain toric varieties remains slope-stable, providing a complete classification for varieties with Picard rank one or two.

Contribution

It offers a comprehensive description of divisors and polarizations ensuring stability of the logarithmic tangent sheaf on toric varieties with Picard rank one or two.

Findings

Characterization of divisors D for stability

Classification of polarizations L for stability

Complete description for Picard rank one or two

Abstract

For an equivariant log pair $(X, D)$ where $X$ is a normal toric variety and $D$ a reduced Weil divisor, we study slope-stability of the logarithmic tangent sheaf $\mathcal{T}_{X}(- \log D)$. We give a complete description of divisors $D$ and polarizations $L$ such that $\mathcal{T}_{X}(- \log D)$ is (semi)stable with respect to $L$ when $X$ has a Picard rank one or two.