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Deformation of moduli spaces of meromorphic $G$-connections on $\mathbb{P}^{1}$ via unfolding of irregular singularities

Kazuki Hiroe

TL;DR

This work develops a comprehensive framework for unfolding unramified irregular singularities of meromorphic $G$-connections on $\mathbb{P}^{1}$ and proves that any nonempty moduli space $\mathcal{M}^{\mathrm{ir}}_{\mathbf{H}}$ deforms to a family whose generic fiber is a moduli space of Fuchsian (regular) $G$-connections. It introduces spectral types and unfolding diagrams to combinatorially encode how local canonical forms decompose under unfolding, and establishes delta-invariant and rigidity-constant deformation theory that tracks these changes across strata. The results yield an obstruction- and existence-aware view of additive Deligne–Simpson problems across unfolding families, and provide an affirmative resolution of Oshima’s conjecture within this framework, including a symplectic/birational interpretation of unfolding maps. Additionally, the paper develops triangular decompositions of truncated orbits to realize explicit unfoldings and to describe the moduli spaces as holomorphic symplectic orbifolds, thereby linking local analytic data to global geometric structures with potential applications to isomonodromic deformations and related integrable systems.

Abstract

Unfolding singular points in linear differential equations is a classical technique for studying the properties of irregular singularities by relating them to regular singularities. In this paper, we propose a general framework for unfolding unramified irregular singularities of meromorphic connections on the trivial principal $G$-bundle over $\mathbb{P}^{1}$. One of our main results is the description of the unfolding of singularities in terms of deformations of their moduli spaces. We show that every moduli space of irreducible meromorphic $G$-connections with unramified irregular singularities on $\mathbb{P}^{1}$ can be deformed into a moduli space of irreducible Fuchsian $G$-connections on $\mathbb{P}^{1}$. Furthermore, we study the unfolding of additive Deligne-Simpson problems, in which the unfolding of irregular singularities naturally generates a family of such problems. As an application of our main result, we prove that a Deligne-Simpson problem for $G$-connections with unramified irregular singularities admits a solution if and only if every unfolded Deligne-Simpson problem in the family admits a simultaneous solution. We also provide a combinatorial and diagrammatic framework of the unfolding process in terms of spectral types and unfolding diagrams. Finally, we address a conjecture proposed by Oshima concerning the existence of irreducible $G$-connections that realize prescribed spectral types and their unfoldings. Our main result gives an affirmative answer to this conjecture.

Deformation of moduli spaces of meromorphic $G$-connections on $\mathbb{P}^{1}$ via unfolding of irregular singularities

TL;DR

This work develops a comprehensive framework for unfolding unramified irregular singularities of meromorphic -connections on and proves that any nonempty moduli space deforms to a family whose generic fiber is a moduli space of Fuchsian (regular) -connections. It introduces spectral types and unfolding diagrams to combinatorially encode how local canonical forms decompose under unfolding, and establishes delta-invariant and rigidity-constant deformation theory that tracks these changes across strata. The results yield an obstruction- and existence-aware view of additive Deligne–Simpson problems across unfolding families, and provide an affirmative resolution of Oshima’s conjecture within this framework, including a symplectic/birational interpretation of unfolding maps. Additionally, the paper develops triangular decompositions of truncated orbits to realize explicit unfoldings and to describe the moduli spaces as holomorphic symplectic orbifolds, thereby linking local analytic data to global geometric structures with potential applications to isomonodromic deformations and related integrable systems.

Abstract

Unfolding singular points in linear differential equations is a classical technique for studying the properties of irregular singularities by relating them to regular singularities. In this paper, we propose a general framework for unfolding unramified irregular singularities of meromorphic connections on the trivial principal -bundle over . One of our main results is the description of the unfolding of singularities in terms of deformations of their moduli spaces. We show that every moduli space of irreducible meromorphic -connections with unramified irregular singularities on can be deformed into a moduli space of irreducible Fuchsian -connections on . Furthermore, we study the unfolding of additive Deligne-Simpson problems, in which the unfolding of irregular singularities naturally generates a family of such problems. As an application of our main result, we prove that a Deligne-Simpson problem for -connections with unramified irregular singularities admits a solution if and only if every unfolded Deligne-Simpson problem in the family admits a simultaneous solution. We also provide a combinatorial and diagrammatic framework of the unfolding process in terms of spectral types and unfolding diagrams. Finally, we address a conjecture proposed by Oshima concerning the existence of irreducible -connections that realize prescribed spectral types and their unfoldings. Our main result gives an affirmative answer to this conjecture.
Paper Structure (56 sections, 58 theorems, 457 equations)

This paper contains 56 sections, 58 theorems, 457 equations.

Table of Contents

  1. Introduction
  2. Classical and recent examples
  3. Unfolding of canonical forms for meromorphic $G$-connections on the formal punctured disk
  4. Statement of the main theorem
  5. Spectral types and unfolding diagrams
  6. Additive Deligne-Simpson problem
  7. A conjecture by Oshima
  8. A further discussion--Unfolding is symplectic and birational
  9. Meromorphic $G$-connections on the formal punctured disk
  10. Hukuhara-Turrittin theory for $\mathfrak{g}$-valued connections
  11. Spectral types of canonical forms
  12. $\delta$-invariants of canonical forms
  13. Truncated orbits of unramified canonical forms
  14. Affine algebraic groups over the ring of formal power series
  15. Truncated orbit as coadjoint orbit
  16. ...and 41 more sections

Key Result

Theorem 1.1

Suppose $\mathcal{M}_{\mathbf{H}}^{\mathrm{ir}}\neq \emptyset$. Then there exists an open neighborhood $\widetilde{\mathbb{B}}_{\mathbf{H}}$ of $\mathbf{0}$ in $\prod_{a\in |D|}\mathbb{C}^{k_{a}+1}$ and a complex orbifold $\mathcal{M}_{\mathbf{H},\widetilde{\mathbb{B}}_{\mathbf{H}}}^{\mathrm{ir}}$ w

Theorems & Definitions (141)

  • Theorem 1.1: Theorem \ref{['thm:main1']}
  • Proposition 1.2: Corollary \ref{['cor:specdecomp']}
  • Theorem 1.5: Theorem \ref{['thm:addDS']}
  • Conjecture 1.6: Conjecture \ref{['conj:osh']}, Oshima Oshi1, Oshi2
  • Theorem 1.7: Theorem \ref{['thm:specdiag']}
  • Corollary 1.8
  • Definition 2.1: Formal meromorphic $G$-connection
  • Definition 2.2: Gauge transformation
  • Definition 2.3: Canonical form
  • Theorem 2.4: Babbitt-Varadarajan BV, cf. Hukukara Huk, Turrittin Tur, and Levelt Lev
  • ...and 131 more