On the stability of tangent bundles on cyclic coverings

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