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On the twistor spaces of ALE gravitational instantons of type $A_{\rm odd}$

Nobuhiro Honda

Abstract

We study the twistor spaces of toric ALE gravitational instantons of type $A_{2n-1}$ and the associated non-standard minitwistor spaces introduced by Hitchin. By analyzing the base locus of the linear system that induces the quotient meromorphic map from the compactified twistor space, we explicitly determine the images of certain distinguished twistor lines as hyperplane sections of the minitwistor space. Using this family of special minitwistor lines as boundary data, we describe the $3$-dimensional family of real minitwistor lines arising from the instanton. The central sphere in the gravitational instanton appears naturally throughout the analysis.

On the twistor spaces of ALE gravitational instantons of type $A_{\rm odd}$

Abstract

We study the twistor spaces of toric ALE gravitational instantons of type and the associated non-standard minitwistor spaces introduced by Hitchin. By analyzing the base locus of the linear system that induces the quotient meromorphic map from the compactified twistor space, we explicitly determine the images of certain distinguished twistor lines as hyperplane sections of the minitwistor space. Using this family of special minitwistor lines as boundary data, we describe the -dimensional family of real minitwistor lines arising from the instanton. The central sphere in the gravitational instanton appears naturally throughout the analysis.
Paper Structure (14 sections, 37 theorems, 114 equations, 10 figures)

This paper contains 14 sections, 37 theorems, 114 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The twistor spaces, their compactifications and minitwistors
  3. A linear system on the compactification
  4. The images of twistor lines to the minitwistor space
  5. Equations of $\mathbb{C}^*$-invariant twistor lines
  6. Elimination of the base locus
  7. Modifications using the nodes
  8. Blowdown to $\widetilde{\mathscr T}\times\mathbb{P}^1$
  9. The images of twistor lines
  10. The complete family of minitwistor lines
  11. The real and the pure imaginary circles of $\Sigma$
  12. Minitwitor lines parameterized by the quarter of $\Sigma$
  13. The complete family of minitwistor lines
  14. Non-real singularities of minitwistor lines

Key Result

Proposition 3.1

For the anti-canonical bundle of $\widetilde{Z}$, we have:

Figures (10)

  • Figure 1: The compactified twistor space $\widetilde{Z}$
  • Figure 2: The resolved minitwistor space $S=\widetilde{\mathscr T}$
  • Figure 3: The elimination of the base locus
  • Figure 4: The blowup of $\bm D\simeq\mathbb{F}_2$
  • Figure 5: The transformations using the nodes.
  • ...and 5 more figures

Theorems & Definitions (41)

  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Proposition 3.8
  • Proposition 3.9
  • Proposition 4.1
  • ...and 31 more