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Geometry of the space of sections of twistor spaces with circle action

Florian Beck, Indranil Biswas, Sebastian Heller, Markus Röser

TL;DR

This work analyzes the geometry of the space $\mathcal{S}$ of holomorphic sections of the Deligne--Hitchin twistor space, focusing on hyperkähler structure, a rotating $S^1$-action, and the associated energy functional. By interpreting Hitchin's meromorphic connection via the Atiyah--Ward transform, the authors show that the residue of the meromorphic connection acts as a holomorphic moment map for the $S^1$ action on $\mathcal{S}$, with fixed points corresponding to $\mathbb{C}^*$-fixed sections and to real variations of Hodge structures. The paper develops an explicit AW transform of $L_Z$ to a line bundle $\mathcal{L}$ on a complexified space $\mathcal{S}^0$, relates it to Hitchin's meromorphic connection, and provides concrete energy and second-variation formulas for $\mathbb{C}^*$-fixed sections, including degree computations of the restricted hyperholomorphic line bundle. Applying this framework to the Deligne--Hitchin moduli space, the authors describe irreducible/admissible sections, prove a global description of the energy as a moment map, and exhibit explicit, nontrivial degrees of $s^*L_Z$ along certain fixed sections, illustrating that the space of sections can be disconnected. The results deepen the twistorial/hyperkähler interpretation of these moduli spaces and yield explicit tools for understanding the HK/QK correspondence via twistorial data.

Abstract

We study the holomorphic symplectic geometry of (the smooth locus of) the space of holomorphic sections of a twistor space with rotating circle action. The twistor space carries a line bundle with a meromorphic connection constructed by Hitchin. We give an interpretation of Hitchin's meromorphic connection in the context of the Atiyah--Ward transform of the corresponding hyperholomorphic line bundle. It is shown that the residue of the meromorphic connection serves as a moment map for the induced circle action, and its critical points are studied. Particular emphasis is given to the example of Deligne--Hitchin moduli spaces.

Geometry of the space of sections of twistor spaces with circle action

TL;DR

This work analyzes the geometry of the space of holomorphic sections of the Deligne--Hitchin twistor space, focusing on hyperkähler structure, a rotating -action, and the associated energy functional. By interpreting Hitchin's meromorphic connection via the Atiyah--Ward transform, the authors show that the residue of the meromorphic connection acts as a holomorphic moment map for the action on , with fixed points corresponding to -fixed sections and to real variations of Hodge structures. The paper develops an explicit AW transform of to a line bundle on a complexified space , relates it to Hitchin's meromorphic connection, and provides concrete energy and second-variation formulas for -fixed sections, including degree computations of the restricted hyperholomorphic line bundle. Applying this framework to the Deligne--Hitchin moduli space, the authors describe irreducible/admissible sections, prove a global description of the energy as a moment map, and exhibit explicit, nontrivial degrees of along certain fixed sections, illustrating that the space of sections can be disconnected. The results deepen the twistorial/hyperkähler interpretation of these moduli spaces and yield explicit tools for understanding the HK/QK correspondence via twistorial data.

Abstract

We study the holomorphic symplectic geometry of (the smooth locus of) the space of holomorphic sections of a twistor space with rotating circle action. The twistor space carries a line bundle with a meromorphic connection constructed by Hitchin. We give an interpretation of Hitchin's meromorphic connection in the context of the Atiyah--Ward transform of the corresponding hyperholomorphic line bundle. It is shown that the residue of the meromorphic connection serves as a moment map for the induced circle action, and its critical points are studied. Particular emphasis is given to the example of Deligne--Hitchin moduli spaces.

Paper Structure

This paper contains 24 sections, 34 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. Geometry of the space of holomorphic sections of a twistor space
  3. Twistor space
  4. Space of holomorphic sections as a complexified hyperKähler manifold
  5. Alternative description of the holomorphic symplectic form $\Omega_x$ on $\mathcal{S}'$
  6. The hyperholomorphic line bundle and the energy functional
  7. Rotating circle actions and the hyperholomorphic line bundle
  8. The $S^1$-action on $\mathcal{S}$ and the energy functional
  9. Critical Points of $\mathcal{E}$
  10. The Atiyah--Ward transform of $L_Z$
  11. The Atiyah--Ward transform and the meromorphic connection
  12. An alternative construction of Hitchin's meromorphic connection
  13. $\tau$-sesquilinear forms and a generalized Chern connection
  14. Space of holomorphic sections of the Deligne--Hitchin moduli space
  15. Hitchin's self-duality equations
  16. ...and 9 more sections

Key Result

Proposition 2.1

Let $Z$ be the twistor space of a hyperKähler manifold $M$ of complex dimension $2d$. Then the set $\mathcal{S}$ of holomorphic sections of is a complex space in a natural way. The tangent space of $s\, \in\, \mathcal{S}$ is $H^0({\mathbb C}P^1,\, N_s)$, and $\mathcal{S}$ is smooth at a point $s\,\in\, \mathcal{S}$ if $H^1({\mathbb C}P^1,\, N_s)\, =\, 0$. If $H^1({\mathbb C}P^1,\, N_s) \,=\, 0$,

Theorems & Definitions (81)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 71 more