Genuinely ramified maps and stable vector bundles

TL;DR

This paper investigates the relationship between stable vector bundles on algebraic varieties connected by a ramified cover, establishing conditions for when bundles on the cover descend to the base and when stability is preserved.

Contribution

It provides a criterion for when a stable vector bundle on the cover arises from a bundle on the base, and proves stability preservation under pullback for certain maps.

Findings

Existence criterion for bundles descending via the pushforward.

Stability preservation under pullback for ramified covers.

Explicit degree and rank conditions for stable 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 $Y$.