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Moduli of Higgs bundles over the two punctured elliptic curve

Thiago Fassarella, Frank Loray

TL;DR

The paper studies SL$_2$ parabolic Higgs bundles on a twice-punctured elliptic curve and describes the full singular fiber structure of the Hitchin fibration. It constructs a degree-two modular map $\Phi$ from the five-punctured sphere to the elliptic case by pullback along the elliptic cover, elementary transformations, and twisting, preserving Hitchin bases and allowing a complete transfer of fiber geometry. Using this map together with known results on the five-punctured sphere, it gives explicit descriptions of all singular Hitchin fibers on the elliptic side, including irreducible nodal, two-component, and the nilpotent cone with nine components, as well as the branch and ramification loci of $\Phi$. The singular locus of the moduli space $\mathcal{H}(C)$ is shown to be irreducible of codimension two, with generic singular elements arising from reducible spectral curves in Hitchin fibers. Altogether, the work links spectral data from the sphere to the elliptic curve, yielding explicit equations and a detailed stratification of Hitchin fibers and the moduli space in this low-genus, two-puncture setting.

Abstract

We study moduli spaces of Higgs bundles with two poles on an elliptic curve. We describe all singular fibers of the Hitchin map, including the nilpotent cone. To achieve this, we consider a modular map that lifts Higgs bundles with five poles on the Riemann sphere to Higgs bundles on the elliptic curve. This map is a two-sheeted covering and we analyze its Galois involution. We prove that the modular map is surjective and determine its ramification locus. In particular, we also obtain an explicit description of the singular locus of the moduli space.

Moduli of Higgs bundles over the two punctured elliptic curve

TL;DR

The paper studies SL parabolic Higgs bundles on a twice-punctured elliptic curve and describes the full singular fiber structure of the Hitchin fibration. It constructs a degree-two modular map from the five-punctured sphere to the elliptic case by pullback along the elliptic cover, elementary transformations, and twisting, preserving Hitchin bases and allowing a complete transfer of fiber geometry. Using this map together with known results on the five-punctured sphere, it gives explicit descriptions of all singular Hitchin fibers on the elliptic side, including irreducible nodal, two-component, and the nilpotent cone with nine components, as well as the branch and ramification loci of . The singular locus of the moduli space is shown to be irreducible of codimension two, with generic singular elements arising from reducible spectral curves in Hitchin fibers. Altogether, the work links spectral data from the sphere to the elliptic curve, yielding explicit equations and a detailed stratification of Hitchin fibers and the moduli space in this low-genus, two-puncture setting.

Abstract

We study moduli spaces of Higgs bundles with two poles on an elliptic curve. We describe all singular fibers of the Hitchin map, including the nilpotent cone. To achieve this, we consider a modular map that lifts Higgs bundles with five poles on the Riemann sphere to Higgs bundles on the elliptic curve. This map is a two-sheeted covering and we analyze its Galois involution. We prove that the modular map is surjective and determine its ramification locus. In particular, we also obtain an explicit description of the singular locus of the moduli space.
Paper Structure (26 sections, 25 theorems, 115 equations, 1 figure, 2 tables)

This paper contains 26 sections, 25 theorems, 115 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Higgs bundles
  4. Twist and elementary transformations
  5. The modular map
  6. The modular degree two covering
  7. Involution of the covering
  8. Recap: Higgs bundles on the five punctured sphere
  9. Higgs fields having unstable parabolic bundles
  10. Unstable parabolic vector bundles
  11. The modular map is a morphism
  12. The $16$ lines of $\mathcal{S}$ go to $8$ lines of $\mathbb{P}^1\times\mathbb{P}^1$
  13. Strictly semistable parabolic vector bundles
  14. The locus of strictly semistable vector bundles
  15. Equation for $C^0$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\det^{-1}(r)$ be a singular fiber of the Hitchin map $\det: \mathcal{H}(C) \to \mathbb{C}^2$ and $Y_r$ be the corresponding spectral curve. Then at least one of the following holds: Moreover, all these cases arise in the moduli space $\mathcal{H}(C)$.

Figures (1)

  • Figure 1: Red points are branch points.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 3.1
  • Proposition 4.1
  • Corollary 4.2
  • Corollary 4.3
  • Proposition 4.4
  • Remark 4.5
  • Proposition 4.6
  • ...and 18 more