On stability of tangent bundle of toric varieties

TL;DR

This paper investigates the stability properties of tangent bundles on nonsingular complex projective toric varieties, providing complete results for certain Fano cases and examples of instability in others using an equivariant approach.

Contribution

It offers a comprehensive analysis of tangent bundle stability on toric varieties, especially Fano varieties of dimension four with low Picard number, and constructs infinite unstable examples.

Findings

Complete stability criteria for Fano toric varieties of dimension four with Picard number ≤ 2

Identification of an infinite set of unstable Fano toric varieties

Application of equivariant methods to analyze tangent bundle stability

Abstract

Let $X$ be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle $T{X}$. In particular, a complete answer is given when $X$ is a Fano toric variety of dimension four with Picard number at most two, complementing earlier work of Nakagawa. We also give an infinite set of examples of Fano toric varieties for which $TX$ is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko and developed further by Perling and Kool.