The H_N filtration of bundles as Frobenius pull-back

TL;DR

This paper investigates the behavior of the Harder-Narasimhan filtration of Frobenius pull-backs of semistable bundles on algebraic curves in characteristic p, providing a counterexample to a previous conjecture.

Contribution

It offers a negative answer to the conjecture that the length of the Harder-Narasimhan filtration of F*W is at most p for semistable bundles W.

Findings

Counterexample to the conjecture about the length of Harder-Narasimhan filtration

Shows that the length can exceed p in certain cases

Provides insight into Frobenius pull-back behavior on bundles

Abstract

Let X be a smooth projective curve over an algebraic closed field of characteristic p and F be the Frobenius morphism of X. Here, I give a negative answer to the guess that the length of the Harder-Narasimhan of F*W is not bigger than p, where W is a semistable bundle on X.