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A Concrete Variant of the Twistor Theorem

Laura Fredrickson, Max Zimet

TL;DR

The paper introduces a concrete variant of the Hitchin–Karlhede–Lindström–Roček twistor theorem by starting from a real manifold $\mathcal{M}$ equipped with a $\mathbb{C}^\times$-family of holomorphic symplectic forms $\varpi(\zeta)$ of the shape $\varpi(\zeta) = - \frac{i}{2\zeta} \omega_+ + \omega_3 - \frac{i}{2} \zeta \omega_-$ with $\omega_- = \bar{\omega}_+$ and $\bar{\omega}_3 = \omega_3$, where $\omega_+$ is holomorphic. The main result proves the existence of a pseudo-hyper-Kähler structure on $\mathcal{M}$ (via a direct real-manifold argument) and shows that if $\varpi(\zeta)$ arises from a pseudo-hyper-Kähler structure, the reconstruction recovers the original metric and complex structures. This yields a more direct and practical route to HK geometries without constructing the twistor space from scratch, and clarifies the role of the $\kappa$-map and the normal-bundle condition in the HKLR framework. The work also connects with Gaiotto–Moore–Neitzke’s program for translating enumerative/semi-flat data into explicit hyper-Kähler metrics near semi-flat limits.

Abstract

In this note, we prove a concrete variant of the twistor theorem of Hitchin--Karlhede--Lindström--Roček which applies when one already has the real manifold on which one wishes to construct a hyper-Kähler structure, and so one does not need to construct it as a parameter space of twistor lines.

A Concrete Variant of the Twistor Theorem

TL;DR

The paper introduces a concrete variant of the Hitchin–Karlhede–Lindström–Roček twistor theorem by starting from a real manifold equipped with a -family of holomorphic symplectic forms of the shape with and , where is holomorphic. The main result proves the existence of a pseudo-hyper-Kähler structure on (via a direct real-manifold argument) and shows that if arises from a pseudo-hyper-Kähler structure, the reconstruction recovers the original metric and complex structures. This yields a more direct and practical route to HK geometries without constructing the twistor space from scratch, and clarifies the role of the -map and the normal-bundle condition in the HKLR framework. The work also connects with Gaiotto–Moore–Neitzke’s program for translating enumerative/semi-flat data into explicit hyper-Kähler metrics near semi-flat limits.

Abstract

In this note, we prove a concrete variant of the twistor theorem of Hitchin--Karlhede--Lindström--Roček which applies when one already has the real manifold on which one wishes to construct a hyper-Kähler structure, and so one does not need to construct it as a parameter space of twistor lines.
Paper Structure (3 sections, 18 theorems, 38 equations, 2 figures)

This paper contains 3 sections, 18 theorems, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. From a family of holomorphic symplectic structures to a pseudo-hyper-Kähler structure

Key Result

Lemma 2.6

The unit quaternion algebra relations in eq:quat are preserved by the natural $SO(3)$ action on $S^2$ (which acts on $\zeta$ via $PSU(2)$ fractional linear transformations).

Figures (2)

  • Figure 1: The twistor sphere.
  • Figure 2: Relevant sections in the Proof of Lemma \ref{['lem:realHol']}.

Theorems & Definitions (39)

  • Definition 2.1
  • Lemma 2.6
  • proof
  • Proposition 2.9
  • proof
  • Corollary 2.13: Corollary to Proposition \ref{['prop:iso0tozeta']}
  • Definition 2.14
  • Corollary 2.15: Corollary to Corollary \ref{['cor:holbundle']}
  • Definition 2.16
  • Proposition 2.17
  • ...and 29 more