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Moduli space of connections on rational irregular curves

Mattia Morbello

TL;DR

The paper studies the moduli space of a class of rank-two irregular connections on the Riemann sphere with a double pole and two simple poles (PV-type connections) and identifies these connections with rational irregular curves together with an accessory parameter. It develops a Deligne–Mumford–style compactification by introducing irregular stable nodal curves and analyzes how an extra parameter hat{p} behaves along the boundary, yielding a 3-dimensional quasi-projective compactification of the PV moduli that is birational to the total space of a P^1-bundle over the Deligne–Mumford compactification of irregular curves. The construction relies on a normal form on E = O(-1) ⊕ O(2), a description of the associated Riccati foliation on the Hirzebruch surface, and a careful analysis of confluence limits of apparent singularities. The resulting compactified space, together with the boundary connections, provides a concrete, birationally tractable framework for studying isomonodromic deformations and the boundary dynamics of the irregular Riemann–Hilbert correspondence, with explicit symmetry actions exchanging poles and residual data. This contributes to a geometric understanding of PV-type phenomena and suggests avenues for future work on boundary foliations and spectral data in irregular isomonodromy.

Abstract

We study the compactification of the moduli space of a certain class of rank-two irregular connections on the Riemann sphere, presenting one double pole and two simple poles. To explicitely build the compactification, we identify a class of such irregular connections with a rational irregular curve and an extra complex parameter. As a first step, we are inspired by Deligne and Mumford's work to compactify the moduli space of rational irregular curves, introducing the notion of irregular stable nodal curve. Second, we will understand the behaviour of the extra complex parameter to conclude the compactification.

Moduli space of connections on rational irregular curves

TL;DR

The paper studies the moduli space of a class of rank-two irregular connections on the Riemann sphere with a double pole and two simple poles (PV-type connections) and identifies these connections with rational irregular curves together with an accessory parameter. It develops a Deligne–Mumford–style compactification by introducing irregular stable nodal curves and analyzes how an extra parameter hat{p} behaves along the boundary, yielding a 3-dimensional quasi-projective compactification of the PV moduli that is birational to the total space of a P^1-bundle over the Deligne–Mumford compactification of irregular curves. The construction relies on a normal form on E = O(-1) ⊕ O(2), a description of the associated Riccati foliation on the Hirzebruch surface, and a careful analysis of confluence limits of apparent singularities. The resulting compactified space, together with the boundary connections, provides a concrete, birationally tractable framework for studying isomonodromic deformations and the boundary dynamics of the irregular Riemann–Hilbert correspondence, with explicit symmetry actions exchanging poles and residual data. This contributes to a geometric understanding of PV-type phenomena and suggests avenues for future work on boundary foliations and spectral data in irregular isomonodromy.

Abstract

We study the compactification of the moduli space of a certain class of rank-two irregular connections on the Riemann sphere, presenting one double pole and two simple poles. To explicitely build the compactification, we identify a class of such irregular connections with a rational irregular curve and an extra complex parameter. As a first step, we are inspired by Deligne and Mumford's work to compactify the moduli space of rational irregular curves, introducing the notion of irregular stable nodal curve. Second, we will understand the behaviour of the extra complex parameter to conclude the compactification.

Paper Structure

This paper contains 17 sections, 35 theorems, 117 equations, 11 figures.

Table of Contents

  1. Preliminary Notions
  2. Linear Meromorphic Connections
  3. Horizontal Sections
  4. Equivalent Connections and Residual Spectral Data
  5. Connections of PV type
  6. Riccati Foliation
  7. Normal Form
  8. Confluence of Singularities
  9. Moduli Space of connections of PV type
  10. Irregular Curves
  11. Compactified Moduli Space of connections of PV type
  12. Deligne-Mumford Compactification of $\mathcal{M}$
  13. Irregular Stable Nodal Curves
  14. Connections on the Boundary
  15. Extension of the line bundle
  16. ...and 2 more sections

Key Result

Theorem A

Let $\Theta=\{\kappa_0,\kappa_1,\kappa_\infty\}$ be generic spectral residual data defined up to integer shifts. The compactified moduli space $\overline\mathfrak{Con}^V_\Theta$ of PV connection with spectral residual data $\Theta$ is a three dimensional quasi projective variety birational to the to

Figures (11)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 6 more figures

Theorems & Definitions (116)

  • Theorem A
  • Definition
  • Remark 1
  • Definition
  • Definition
  • Example 1: Euler System
  • Proposition 1.1
  • Remark 2
  • Definition
  • Definition
  • ...and 106 more