Hitchin-Mochizuki morphism, Opers and Frobenius-destabilized vector bundles over curves

TL;DR

This paper constructs an explicit atlas for Frobenius-destabilized bundles over curves in large characteristic, establishes a correspondence with opers of zero p-curvature, and analyzes the finiteness and dimensions of related moduli spaces.

Contribution

It provides a new explicit construction of Frobenius-destabilized bundles, generalizes Mochizuki's results to higher ranks, and links these bundles to opers with zero p-curvature.

Findings

Constructed an atlas for Frobenius-destabilized bundles using Quot-schemes.

Established a bijection between certain stable bundles and opers with zero p-curvature.

Proved finiteness of these sets and computed dimensions of related moduli spaces.

Abstract

Let X be a smooth projective curve of genus g \textgreater{}1 defined over an algebraically closed field k of characteristic p \textgreater{}0. For p sufficiently large (explicitly given in terms of r,g) we construct an atlas for the locus of all Frobenius-destabilized bundles (i.e. we construct all Frobenius-destabilized bundles of degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E over X such that the pull-back F^*(E) under the Frobenius morphism of X has maximal Harder-Narasimhan polygon and the set of opers having zero p-curvature. We also show that, after fixing the determinant, these sets are finite, which enables us to derive the dimension of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X. The finiteness is proved by studying the properties of the Hitchin-Mochizuki morphism; an alternative approach to finiteness has been realized in a recent preprint by Chen and Zhu. In particular we prove a generalization of a result of Mochizuki to higher ranks.