On the stability of tangent bundles on cyclic coverings

TL;DR

This paper investigates the stability of tangent bundles on cyclic coverings of smooth projective surfaces, establishing conditions under which the tangent bundle of the covering remains semi-stable if the base surface's tangent bundle is semi-stable.

Contribution

It provides new criteria for the semi-stability of tangent bundles on cyclic coverings based on the semi-stability of the base surface's tangent bundle.

Findings

Tangent bundle of cyclic coverings can be semi-stable under certain conditions.

Semi-stability of the tangent bundle on the base surface implies semi-stability on the covering.

Results depend on the characteristic of the field and the properties of the branch divisor.

Abstract

Let $Y$ be a smooth projective surface defined over an algebraically closed field $k$ with ${\rm Char}\ k\nmid n$, and let $\pi:X\rightarrow Y$ be a $n$-cyclic covering branched along a smooth divisor $B$. We show that under some conditions $\mathscr{T}_X$ is semi-stable with respect to $\pi^*\mathcal {H}$ if the tangent bundle $\mathscr{T}_Y$ is semi-stable with respect to some ample line bundle $\mathcal {H}$ on $Y$.