Strongly semistable reduction of syzygy bundles on plane curves

TL;DR

This paper studies degenerations of syzygy bundles on plane curves over p-adic fields, showing how certain models lead to bundles with strongly semistable reduction, contributing to p-adic Simpson theory and local systems.

Contribution

It introduces a method using Mustafin varieties to construct models of plane curves with special degenerations, demonstrating strongly semistable reduction of syzygy bundles in this context.

Findings

Constructed models with special degenerations of plane curves.

Proved syzygy bundles have strongly semistable reduction on these models.

Applied methods to Fermat curve example, showing potential strongly semistable reduction.

Abstract

We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamental open problem in $p$-adic Simpson theory to determine the category of Higgs bundles corresponding to continuous representations of the \'etale fundamental group of a curve. Faltings' $p$-adic Simpson correspondence and work of Deninger and the second author shows that bundles with Higgs field zero and potentially strongly semistable reduction fall into this category. Hence the results in the present paper determine a class of syzygy bundles on plane curves giving rise to a $p$-adic local system. We apply our methods to a concrete example on the Fermat curve suggested by Brenner and prove that this bundle has potentially strongly semistable reduction.