Direct image of structure sheaf and parabolic stability

TL;DR

This paper proves that a naturally constructed parabolic vector bundle from a morphism between smooth projective curves is parabolic polystable, and identifies conditions under which it is actually parabolic stable.

Contribution

It establishes the parabolic polystability of the direct image of the structure sheaf with a natural parabolic structure and provides criteria for its stability.

Findings

The bundle $(f_*{ m O}_X)/{ m O}_Y$ is parabolic polystable.

Conditions are identified that guarantee parabolic stability.

The results depend on the characteristic of the base field and the morphism's properties.

Abstract

Let $f : X \rightarrow Y$ be a dominant generically smooth morphism between irreducible smooth projective curves over an algebraically closed field $k$ such that ${\rm Char}(k)> \text{degree}(f)$ if the characteristic of $k$ is nonzero. We prove that $(f_*{\mathcal O}_X)/{\mathcal O}_Y$ equipped with a natural parabolic structure is parabolic polystable. Several conditions are given that ensure that the parabolic vector bundle $(f_*{\mathcal O}_X)/{\mathcal O}_Y$ is actually parabolic stable.