Genuinely ramified maps and stable vector bundles

Abstract

Let $f:X\to Y$ be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between étale fundamental groups $f_{\ast }:{\pi }_1^{\mathrm{et}}(X)\to {\pi }_1^{\mathrm{et}}(Y)$ 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 $f^{\ast }W$ isomorphic to $E$ if and only if the direct image $f_{\ast }E$ contains a stable vector bundle $F$ such that $$\frac{\mathrm{degree}(F)}{\mathrm{rank}(F)}=\frac{1}{\mathrm{degree}(f)}\cdot \frac{\mathrm{degree}(E)}{\mathrm{rank}(E)}$$ We also prove that $f^{\ast }V$ is stable for every stable vector bundle $V$ on $Y$.

Final version