Moduli of unramified irregular singular parabolic connections on a smooth projective curve

TL;DR

This paper constructs and analyzes a moduli space of unramified irregular singular parabolic connections on smooth projective curves, demonstrating its smoothness, symplectic structure, and a Riemann-Hilbert correspondence linking it to monodromy data, with implications for isomonodromic deformations.

Contribution

It introduces a new moduli scheme for unramified irregular singular parabolic connections and establishes a symplectic structure and a Riemann-Hilbert correspondence for these objects.

Findings

The moduli space is smooth and symplectic.

The Riemann-Hilbert correspondence is an analytic isomorphism for generic exponents.

Differential systems from isomonodromic deformations have the geometric Painlevé property.

Abstract

In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann-Hilbert correspondence from the moduli space of the connections to the moduli space of the associated generalized monodromy data gives an analytic isomorphism. This shows that differential systems arising from (generalized) isomonodromic deformations of corresponding unramified irregular singular parabolic connections admit geometric Painlev\'e property as in the regular singular cases proved generally in [8].