Semi-stable vector bundles on fibred varieties

Let $\pi :Y\to X$ be a surjective morphism between two irreducible, smooth complex projective varieties with $\mathrm{d}\mathrm{i}\mathrm{m}Y>\mathrm{d}\mathrm{i}\mathrm{m}X>0$. We consider polarizations of the form $L_c=L+c\cdot {\pi }^{\ast }A$ on $Y$, with $c>0$, where $L,A$ are ample line bundles on $Y,X$ respectively. For $c$ sufficiently large, we show that the restriction of a torsion free sheaf $\mathcal{F}$ on $Y$ to the generic fibre $\mathrm{\Phi }$ of $\pi$ is semi-stable as soon as $\mathcal{F}$ is $L_c$-semi-stable; conversely, if $\mathcal{F}\otimes {\mathcal{O}}_{\mathrm{\Phi }}$ is $L$-stable on $\mathrm{\Phi }$, then $\mathcal{F}$ is $L_c$-stable. We obtain explicit lower bounds for $c$ satisfying these properties. Using this result, we discuss the construction of semi-stable vector bundles on Hirzebruch surfaces and on ${\mathbb{P}}^2$-bundles over ${\mathbb{P}}^1$, and establish the irreducibility and the rationality of the corresponding moduli spaces.