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Hypercomplex structures arising from twistor spaces

Shuo Wang, Bin Xu

Abstract

A hyperkähler manifold is defined as a Riemannian manifold endowed with three covariantly constant complex structures that are quaternionically related. A twistor space is characterized as a holomorphic fiber bundle $p: \mathcal{Z} \rightarrow \mathbb{CP}^1$ possesses properties such as a family of holomorphic sections whose normal bundle is $\bigoplus^{2n}\mathcal{O}(1)$, a holomorphic section of $Λ^2(N\mathcal{Z})\otimes p^*(\mathcal{O}(2))$ that defines a symplectic form on each fiber, and a compatible real structure. According to the Hitchin-Karlhede-Lindström-Roček theorem (Comm. Math. Phys., 108(4):535-589, 1987), there exists a hyperkähler metric on the parameter space $M$ for the real sections of $\mathcal{Z}$. Utilizing the Kodaira-Spencer deformation theory, we facilitate the construction of a hypercomplex structure on $M$, predicated upon more relaxed presuppositions concerning $\mathcal{Z}$. This effort enriches our understanding of the classical theorem by Hitchin-Karlhede-Lindström-Roček.

Hypercomplex structures arising from twistor spaces

Abstract

A hyperkähler manifold is defined as a Riemannian manifold endowed with three covariantly constant complex structures that are quaternionically related. A twistor space is characterized as a holomorphic fiber bundle possesses properties such as a family of holomorphic sections whose normal bundle is , a holomorphic section of that defines a symplectic form on each fiber, and a compatible real structure. According to the Hitchin-Karlhede-Lindström-Roček theorem (Comm. Math. Phys., 108(4):535-589, 1987), there exists a hyperkähler metric on the parameter space for the real sections of . Utilizing the Kodaira-Spencer deformation theory, we facilitate the construction of a hypercomplex structure on , predicated upon more relaxed presuppositions concerning . This effort enriches our understanding of the classical theorem by Hitchin-Karlhede-Lindström-Roček.
Paper Structure (5 sections, 28 theorems, 112 equations, 4 figures)

This paper contains 5 sections, 28 theorems, 112 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Deformation of compact complex submanifolds
  3. real structures on $\bigoplus^{2n}\mathcal{O}(1)\rightarrow\mathbb{CP}^1$
  4. Twistor spaces
  5. the twistor space of $\mathbb{C}^{2n}$

Key Result

Theorem 1.1

hitchin1987hyperkahler Let $\mathcal{Z}$ be a complex manifold and $\sigma$ the antipodal map over $\mathbb{CP}^1$. Assuming that conditions (1), (2), (3) and (4) are satisfied, and let $M$ be the set of real holomorphic sections in $\mathcal{M}$. If $M \neq \emptyset$, then it constitutes a smooth

Figures (4)

  • Figure 1: System of canonical local charts
  • Figure 2: Deformation of compact complex submanifolds
  • Figure 3: Canonical local deformation
  • Figure 4: Deformation space

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 59 more