Isomonodromic deformations of connections with singularities of parahoric formal type

TL;DR

This paper develops a geometric framework for studying connections with irregular singularities of parahoric formal type, constructing their moduli space and deriving associated isomonodromy equations as an integrable system.

Contribution

It generalizes existing theories to include parahoric formal types and explicitly computes the isomonodromy equations for these connections.

Findings

Constructed the moduli space as a smooth Poisson reduction.

Derived explicit isomonodromy equations as an integrable system.

Extended classical results to connections with parahoric singularities.

Abstract

In previous work, the authors have developed a geometric theory of fundamental strata to study connections on the projective line with irregular singularities of parahoric formal type. In this paper, the moduli space of connections that contain regular fundamental strata with fixed combinatorics at each singular point is constructed as a smooth Poisson reduction. The authors then explicitly compute the isomonodromy equations as an integrable system. This result generalizes work of Jimbo, Miwa, and Ueno to connections whose singularities have parahoric formal type.