On the stability of pulled back parabolic vector bundles

TL;DR

This paper proves that the pullback of a stable parabolic vector bundle remains stable under certain conditions when pulled back via a nonconstant map from a smooth projective curve.

Contribution

It establishes the stability of pulled back parabolic vector bundles in the case where a specific associated subbundle has rank one.

Findings

Pullback of stable parabolic bundles remains stable when rank of associated subbundle is one.

Constructs a natural subbundle using data from points and integers on the curve.

Provides conditions under which stability is preserved under pullback.

Abstract

Take an irreducible smooth projective curve $X$ defined over an algebraically closed field of characteristic zero, and fix finitely many distinct point $D\, =\, \{x_1,\, \cdots,\, x_n\}$ of it; for each point $x\, \in\, D$ fix a positive integer $N_x$. Take a nonconstant map $f\, :\, Y\, \longrightarrow \, X$ from an irreducible smooth projective curve. We construct a natural subbundle $\mathcal{F}\, \subset\, f_*{\mathcal O}_Y$ using $(D,\, \{N_x\}_{x\in D})$. Let $E_*$ be a stable parabolic vector bundle whose parabolic weights at each $x\, \in\, D$ are integral multiples of $\frac{1}{N_x}$. We prove that the pullback $f^*E_*$ is also parabolic stable, if ${\rm rank}(\mathcal{F})\,=\, 1$.