Harder–Narasimhan filtration of the bundles as Frobenius pull-back

Abstract

Let X be a smooth projective curve of genus \({g \geq 2}\) over an algebraically closed field k of characteristic \({p > 0}\). Let \({F_{X/k} : X \rightarrow X_{1}}\) be the relative Frobenius morphism, and E be a semistable vector bundle on X. Mehta and Pauly asked that whether the length of the Harder–Narasimhan filtration of \({(F_{X/k})^*E}\) is at most p. In this article, we answer the above question negatively by constructing an example.