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From $λ$-connections to $PSL_2(\mathbb{C})$-opers with apparent singularities

Duong Dinh

TL;DR

This work develops a comprehensive framework to pass from rank-2 ${SL}_2({\mathbb C})$ $\lambda$-connections, via sub-line bundles, to ${PSL}_2({\mathbb C})$-opers with prescribed apparent singularities on a genus $g>1$ surface. It constructs an affine-symplectic setup ${\mathcal M}^d_\lambda$ modeled over $T^*{\mathcal N_d}$ and defines a generalized Separation of Variables map ${\mathrm SoV_{\lambda}}$ to the moduli of opers with $m=2g-2-2d$ apparent singularities, proving this map is dominant and Poisson. The paper then characterizes opers through positions and residue parameters of apparent singularities, establishes a robust moduli theory with a natural Poisson structure, and analyzes the inverse construction from opers back to triples $(E,L,\nabla_\lambda)$. It also connects these constructions to wobbly bundles, Lagrangian leaves, and the conformal limit linking Hitchin, Higgs, and de Rham moduli spaces, highlighting potential deformations and how fixed divisors induce Lagrangian leaves. Overall, the results generalize earlier SoV constructions for Higgs bundles to the $\lambda$-connection setting and provide a geometrically rich, Poisson-compatible pathway between spectral data, opers, and monodromy representations in higher-genus contexts.

Abstract

On a Riemann surface of genus $> 1$, we discuss how to construct opers with apparent singularities from $SL_2(\mathbb{C})$ $λ$-connections $(E, \nabla_λ)$ and sub-line bundles $L$ of $E$. This construction defines a rational map from a space which captures important data of triples $(E, L, \nabla_λ)$ to a space which parametrises the positions and residue parameters of the induced apparent singularities. We show that this is a Poisson map with respect to natural Poisson structures. The relations to wobbly bundles and Lagrangians in the moduli spaces of Higgs bundles and $λ$-connections are discussed.

From $λ$-connections to $PSL_2(\mathbb{C})$-opers with apparent singularities

TL;DR

This work develops a comprehensive framework to pass from rank-2 -connections, via sub-line bundles, to -opers with prescribed apparent singularities on a genus surface. It constructs an affine-symplectic setup modeled over and defines a generalized Separation of Variables map to the moduli of opers with apparent singularities, proving this map is dominant and Poisson. The paper then characterizes opers through positions and residue parameters of apparent singularities, establishes a robust moduli theory with a natural Poisson structure, and analyzes the inverse construction from opers back to triples . It also connects these constructions to wobbly bundles, Lagrangian leaves, and the conformal limit linking Hitchin, Higgs, and de Rham moduli spaces, highlighting potential deformations and how fixed divisors induce Lagrangian leaves. Overall, the results generalize earlier SoV constructions for Higgs bundles to the -connection setting and provide a geometrically rich, Poisson-compatible pathway between spectral data, opers, and monodromy representations in higher-genus contexts.

Abstract

On a Riemann surface of genus , we discuss how to construct opers with apparent singularities from -connections and sub-line bundles of . This construction defines a rational map from a space which captures important data of triples to a space which parametrises the positions and residue parameters of the induced apparent singularities. We show that this is a Poisson map with respect to natural Poisson structures. The relations to wobbly bundles and Lagrangians in the moduli spaces of Higgs bundles and -connections are discussed.

Paper Structure

This paper contains 45 sections, 25 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the paper
  4. Rank-2 $\lambda$-connections and sub-line bundles
  5. Moduli space of pairs $(E, L)$ and its cotangent spaces
  6. Coordinates on ${\mathcal{M}}_{d}$
  7. Higgs fields in terms of meromorphic differentials
  8. Symplectic structures on $T^\ast {\mathcal{M}}_{d}$ and $T^\ast \mathcal{N}_{d}$
  9. Fibers of $\mathcal{N}_{d} \dashrightarrow \mathcal{N}$
  10. $\lambda$-connections in terms of Abelian differentials
  11. Symplectic affine bundles on $\mathcal{N}_{d}$ and ${\mathcal{M}}_{d}$
  12. Affine bundle on ${\mathcal{M}}_{d}$
  13. Symplectic structure
  14. Symplectic affine bundle on $\mathcal{N}_{d}$
  15. Maps from ${\mathcal{M}}^s_\lambda$
  16. ...and 30 more sections

Key Result

Theorem 1.1

DT1${\mathrm{SoV}}$ is a dominant Poisson map with respect to the natural Poisson structure of cotangent bundles.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • ...and 33 more