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ALE spaces and nodal curves

Nigel Hitchin

Abstract

We consider the twistor theory approach to Kronheimer's ALE metrics on resolutions of the quotient of C^2 by a finite subgroup of SU(2). The circle action on the 4-manifold induces a C^* action on a compactification of the twistor space and we identify the orbit of a generic twistor line as a nodal rational curve in a particular cohomology class of a projective rational surface. Using the results of N.Honda et al we identify this surface with the minitwistor space for the Einstein-Weyl structure on the 3-dimensional quotient of the ALE space by the circle action.

ALE spaces and nodal curves

Abstract

We consider the twistor theory approach to Kronheimer's ALE metrics on resolutions of the quotient of C^2 by a finite subgroup of SU(2). The circle action on the 4-manifold induces a C^* action on a compactification of the twistor space and we identify the orbit of a generic twistor line as a nodal rational curve in a particular cohomology class of a projective rational surface. Using the results of N.Honda et al we identify this surface with the minitwistor space for the Einstein-Weyl structure on the 3-dimensional quotient of the ALE space by the circle action.
Paper Structure (9 sections, 1 theorem, 12 equations, 2 figures)

This paper contains 9 sections, 1 theorem, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The twistor construction
  3. Compactifications
  4. The $\mathbf{C}^*$-action
  5. Examples
  6. ${\bf A_{2\ell-1}}$
  7. ${\bf D_4}$
  8. The central sphere
  9. Einstein-Weyl manifolds

Key Result

Theorem 1

The divisor class $Q$ of the image of a twistor line in the rational surface $\bar{Z}_1$ is given by and in each case the intersection number $Q^2=\vert \Gamma\vert$ and $KQ=-4$ where the ALE space is asymptotic to $\mathbf{C}^2/\Gamma$.

Figures (2)

  • Figure 1: $D_4$ blowing up
  • Figure 2: the planar model

Theorems & Definitions (1)

  • Theorem 1