Constructing Parabolic Non-Abelian Hodge Correspondence in Positive Characteristic Using Parabolic Bases

TL;DR

This paper develops a localized framework using parabolic bases to construct the parabolic non-abelian Hodge correspondence in positive characteristic, extending prior work and introducing new algorithms for parabolic Higgs-de Rham flows.

Contribution

It introduces parabolic bases for localized analysis and proposes a new method for the parabolic non-abelian Hodge correspondence in positive characteristic, extending existing theories.

Findings

Established a localized framework for parabolic bundles.

Extended the non-abelian Hodge correspondence to positive characteristic.

Modified the Sun-Yang-Zuo algorithm for parabolic Higgs-de Rham flows.

Abstract

We introduce the concept of parabolic bases to establish a localized framework for parabolic bundles and parabolic $\lambda$-connections. Building on this foundation, we propose a novel method for constructing the parabolic non-abelian Hodge correspondence in positive characteristic, extending the work originally developed by Krishnamoorthy and Sheng for algebraic curves. Additionally, we investigate the rank $2$ parabolic Higgs-de Rham flow operator and present a modified version of the Sun-Yang-Zuo algorithm, specifically adapted to the parabolic setting.