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Principal twistor models and asymptotic hyperkähler metrics

Ryota Kotani

Abstract

Let $X$ be a conical symplectic variety admitting a crepant resolution $Y$. Based on the theory of universal Poisson deformations, we construct a complex manifold called the principal twistor model associated with $Y$. We prove a universality theorem for this model: if the regular locus of $X$ admits a hyperkähler cone metric, then the twistor space of any algebraic hyperkähler metric on $Y$ asymptotic to this cone metric is uniquely recovered by slicing the principal twistor model. As an application, we use this universality to study the moduli space of hyperkähler structures with asymptotic behavior, and show that it admits an inclusion into a finite-dimensional real vector space.

Principal twistor models and asymptotic hyperkähler metrics

Abstract

Let be a conical symplectic variety admitting a crepant resolution . Based on the theory of universal Poisson deformations, we construct a complex manifold called the principal twistor model associated with . We prove a universality theorem for this model: if the regular locus of admits a hyperkähler cone metric, then the twistor space of any algebraic hyperkähler metric on asymptotic to this cone metric is uniquely recovered by slicing the principal twistor model. As an application, we use this universality to study the moduli space of hyperkähler structures with asymptotic behavior, and show that it admits an inclusion into a finite-dimensional real vector space.
Paper Structure (25 sections, 40 theorems, 50 equations, 2 figures)

This paper contains 25 sections, 40 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Poisson deformations of conical symplectic varieties with good triples
  3. Definitions
  4. Crepant resolutions
  5. Universal Poisson deformations
  6. Principal twistor model
  7. Definitions
  8. $\mathbb{C}^*$-gluing construction
  9. Definition
  10. Extension of $\mathbb{C}^*$-action
  11. Extension of quaternionic structure
  12. Extension of holomorphic differential forms
  13. Construction of the principal twistor model
  14. Universality of the Principal Twistor Model
  15. Twistor Cones and Hyperkähler Cone Metrics
  16. ...and 10 more sections

Key Result

Proposition 1.1

Let $X$ be a conical symplectic variety with a good triple, and assume that it admits a crepant resolution $Y$. Let $\mathcal{Y} \rightarrow \mathcal{C}$ denote the universal Poisson deformation of $Y$, where the base space $\mathcal{C}$ is isomorphic to $H^2(Y;\mathbb{C})$. Then, the good triple na

Figures (2)

  • Figure 1: A conceptual diagram of the model $\mathcal{Y}(1)$, where the three coordinate axes correspond to $\mathcal{C}$, $Y$, and $\mathbb{P}^1$. The three faces correspond to $\mathcal{C}(2)$, $\mathcal{Y}$, and $Y(1)$.
  • Figure 2: A conceptual diagram showing the twistor model $Z_s$ associated with a real section $s$ embedded inside the model $\mathcal{Y}(1)$.

Theorems & Definitions (121)

  • Proposition 1.1: Proposition \ref{['main prop: PTM of Y']}
  • Remark 1.2
  • Theorem 1.3: Theorem \ref{['main thm: universality of PTM']}
  • Remark 1.4
  • Corollary 1.5: Cor. \ref{['cor:inclusion of moduli space']}, Cor. \ref{['cor:moduli of HK str, isolated sing case']}
  • Remark 1.6
  • Definition 2.1: cf. Namikawa15
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 111 more