Genuinely ramified maps and stable vector bundles

Abstract: Let $f : X \rightarrow Y$ be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between \'etale fundamental groups $f_* : \pi_1^{\rm et}(X)\rightarrow\pi_1^{\rm et}(Y)$ is surjective. Fix a polarization on $Y$ and equip $X$ with the pullback, by $f$, of this polarization on $Y$. Given a stable vector bundle $E$ on $X$, we prove that there is a vector bundle $W$ on $Y$ with $f^*W$ isomorphic to $E$ if and only if the direct image $f_*E$ contains a stable vector bundle $F$ such that $$ \frac{{\rm degree}(F)}{{\rm rank}(F)}= \frac{1}{{\rm degree}(f)}\cdot \frac{{\rm degree}(E)}{{\rm rank}(E)} $$ We also prove that $f^*V$ is stable for every stable vector bundle $V$ on $Y$.