Parahoric reduction theory of formal connections (or Higgs fields)

TL;DR

This paper develops a comprehensive parahoric reduction theory for formal connections and Higgs fields, extending previous results and providing new criteria and equivalences for regular singularities.

Contribution

It generalizes existing reduction theories to include parahoric structures and establishes new criteria and equivalences for regular singularities in formal connections.

Findings

Proves the equivalence between extrinsic and intrinsic definitions of regular singularity.

Provides a criterion for relative regularity of formal connections.

Demonstrates a parahoric version of Frenkel-Zhu's Borel reduction theorem.

Abstract

In this paper, we establish the parahoric reduction theory of formal connections (or Higgs fields) on a formal principal bundle with parahoric structures, which generalizes Babbitt-Varadarajan's result for the case without parahoric structures [5] and Boalch's result for the case of regular singularity [9]. As applications, we prove the equivalence between extrinsic definition and intrinsic definition of regular singularity and provide a criterion of relative regularity for formal connections, and also demonstrate a parahoric version of Frenkel-Zhu's Borel reduction theorem of formal connections [23].