On a p-adic version of Narasimhan and Seshadri's theorem

TL;DR

This paper investigates p-adic analogues of a classical theorem on vector bundles over curves, establishing conditions under which certain types of semistability are equivalent, and providing a negative answer to a question posed by Faltings.

Contribution

It proves that for curves with good reduction and certain p, the conditions of potentially strongly semistable reduction and strongly semistable reduction are equivalent, addressing Faltings' question.

Findings

Equivalence of semistability conditions under specified conditions

Negative answer to Faltings' question on p-adic Simpson correspondence

Conditions for vector bundles with stable reduction

Abstract

Consider a smooth projective curve C of genus g over a complete discrete valuation field of characteristic 0 and residue field \Fbar_p. Motivated by Narasimhan and Seshadri's theorem, Faltings asked whether all semistable vector bundles of degree 0 over C_{\C_p} are in the image of the p-adic Simpson correspondence. Works of Deninger-Werner and Xu show that this is equivalent for the vector bundle to having potentially strongly semistable reduction. We prove that if C has good reduction, p>r(r-1) (g-1) and we consider a vector bundle of rank r with stable reduction, the conditions of having potentially strongly semistable reduction and of having strongly semistable reduction are equivalent. In particular, we provide a negative answer to Faltings' question