Ramified covering maps and stability of pulled back bundles

TL;DR

This paper characterizes when the pullback of a stable vector bundle under a morphism between curves remains stable, establishing a precise condition involving the ramification properties of the morphism.

Contribution

It provides a necessary and sufficient condition, called genuine ramification, for the stability of pulled back bundles on algebraic curves.

Findings

Pullback of stable bundles remains stable iff the morphism is genuinely ramified.

Genuine ramification is characterized by the maximal semistable subbundle condition.

The result links ramification properties with stability preservation in vector bundles.

Abstract

Let $f:C\rightarrow D$ be a nonconstant separable morphism between irreducible smooth projective curves defined over an algebraically closed field. We say that $f$ is genuinely ramified if ${\mathcal O}_D$ is the maximal semistable subbundle of $f_*{\mathcal O}_C$ (equivalently, the homomorphism of etale fundamental groups is surjective). We prove that the pullback $f^*E\rightarrow C$ is stable for every stable vector bundle $E$ on $D$ if and only if $f$ is genuinely ramified.