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Hyperelliptic curves, minitwistors, and spacelike Zoll spaces

Nobuhiro Honda

TL;DR

The paper constructs compact minitwistor spaces from real hyperelliptic curves of genus $g>1$ and uses them to generate a family of Lorentzian Einstein-Weyl structures on 3-manifolds diffeomorphic to $\mathbb{S}^2\times I$, all real analytic and admitting an $\mathbb{S}^1$-action. It shows every EW space in the family is spacelike Zoll (all spacelike geodesics are closed and simple) and that the moduli space of these EW structures has dimension $2g-1$, mirroring the complex-structure moduli of the underlying hyperelliptic curves. The construction hinges on a detailed real Abel–Jacobi analysis, Seifert surfaces in the Jacobian, and a nuanced completion of the minitwistor spaces by incorporating irregular minitwistor lines; the results tie into Hitchin’s work on ALE instantons and reveal a close relation to the A-type instanton settings. Overall, the work extends EW geometry from genus one to higher genus, providing a rich link between hyperelliptic geometry, minitwistor theory, and Lorentzian Zoll phenomena with explicit moduli and automorphism structures.

Abstract

We construct a compact minitwistor space from a hyperelliptic curve with real structure and show that it yields a lot of new Lorentzian Einstein-Weyl spaces all of which are diffeomorphic to the 3-dimensional deSitter space. These structures are real analytic, admit a circle symmetry and moreover, all their spacelike geodesics are closed and simple. The number of the nodes of minitwistor lines on the minitwistor space is equal to the genus of the hyperelliptic curve and is taken arbitrarily. These Einstein-Weyl structures deform as the hyperelliptic curves deform, and so have $(2g-1)$-dimensional moduli space, where $g$ is the genus of the hyperelliptic curve. A relationship between the minitwistor spaces recently obtained by Hitchin from ALE gravitational instantons is also given for A$_{\rm odd}$-type.

Hyperelliptic curves, minitwistors, and spacelike Zoll spaces

TL;DR

The paper constructs compact minitwistor spaces from real hyperelliptic curves of genus and uses them to generate a family of Lorentzian Einstein-Weyl structures on 3-manifolds diffeomorphic to , all real analytic and admitting an -action. It shows every EW space in the family is spacelike Zoll (all spacelike geodesics are closed and simple) and that the moduli space of these EW structures has dimension , mirroring the complex-structure moduli of the underlying hyperelliptic curves. The construction hinges on a detailed real Abel–Jacobi analysis, Seifert surfaces in the Jacobian, and a nuanced completion of the minitwistor spaces by incorporating irregular minitwistor lines; the results tie into Hitchin’s work on ALE instantons and reveal a close relation to the A-type instanton settings. Overall, the work extends EW geometry from genus one to higher genus, providing a rich link between hyperelliptic geometry, minitwistor theory, and Lorentzian Zoll phenomena with explicit moduli and automorphism structures.

Abstract

We construct a compact minitwistor space from a hyperelliptic curve with real structure and show that it yields a lot of new Lorentzian Einstein-Weyl spaces all of which are diffeomorphic to the 3-dimensional deSitter space. These structures are real analytic, admit a circle symmetry and moreover, all their spacelike geodesics are closed and simple. The number of the nodes of minitwistor lines on the minitwistor space is equal to the genus of the hyperelliptic curve and is taken arbitrarily. These Einstein-Weyl structures deform as the hyperelliptic curves deform, and so have -dimensional moduli space, where is the genus of the hyperelliptic curve. A relationship between the minitwistor spaces recently obtained by Hitchin from ALE gravitational instantons is also given for A-type.

Paper Structure

This paper contains 19 sections, 46 theorems, 107 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Hyperelliptic curves and their tangent hyperplanes
  3. The Jacobian varieties of hyperelliptic curves with real structures
  4. Some subsets in the real locus of the Jacobian variety
  5. Seifert surfaces in the real form of the Jacobian varieties
  6. The minitwistor spaces and their minitwistor lines
  7. The minitwistor spaces
  8. Regular minitwistor lines
  9. Irreguglar minitwistor lines
  10. The completion of $M_k^{\circ}$
  11. Properties of the Einstein-Weyl structures
  12. Extension of the EW structures
  13. Spacelike Zoll property of the EW structures.
  14. Automorphisms and the moduli spaces of the Einstein-Weyl structures
  15. The invariance of $\bm I_k\cup\bm I'_k$.
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $g>1$, there exist compact minitwistor spaces of genus $g$ equipped with real structures whose real EW spaces satisfy the following properties:

Figures (10)

  • Figure 1: The 1-cycles in $\Sigma$
  • Figure 2: The region $W_{g+1,\,g+1}$
  • Figure 3: The Seifert surface $\mathfrak S_k$ and its boundary $\mathscr S^1_k \cup \widehat{\mathscr S}^1_k$
  • Figure 4: The disks in the cone ${\rm C}(\Lambda)$ in the case $g=2$.
  • Figure 5: Obtaining $M_k^{\circ}$ from the Seifert surface $\mathfrak S_k$
  • ...and 5 more figures

Theorems & Definitions (58)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 48 more