On a p-adic version of Narasimhan and Seshadri's theorem
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
Keywords
Cite
@article{arxiv.2406.12766,
title = {On a p-adic version of Narasimhan and Seshadri's theorem},
author = {Fabrizio Andreatta},
journal= {arXiv preprint arXiv:2406.12766},
year = {2025}
}
Comments
Using the paper "Higgs bundles over the good reduction of a quaternionic Shimura curve" by Mao Sheng, Jiajin Zhang, Kang Zuo, one can constrct a counterexample to the main claim of the paper