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On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities

Mengwei Hu

Abstract

Kleinian singularities are quotients of $\mathbb{C}^2$ by finite subgroups of $\mathrm{SL}_2(\mathbb{C})$. They are in bijection with the simply-laced Dynkin diagrams via the McKay correspondence. Anti-Poisson involutions and their fixed point loci appear naturally when we want to classify irreducible Harish-Chandra modules over Kleinian singularities. There are three goals of this paper. The first is to classify anti-Poisson involutions of Kleinian singularities up to conjugation by graded Poisson automorphisms. The second is to describe the scheme-theoretic fixed point loci of Kleinian singularities under anti-Poisson involutions. The last and the main goal is to describe the scheme-theoretic preimages of the fixed point loci under minimal resolutions of Kleinian singularities, which are singular Lagrangian subvarieties in the minimal resolutions whose irreducible components are $\mathbb{P}^1$'s and $\mathbb{A}^1$'s.

On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities

Abstract

Kleinian singularities are quotients of by finite subgroups of . They are in bijection with the simply-laced Dynkin diagrams via the McKay correspondence. Anti-Poisson involutions and their fixed point loci appear naturally when we want to classify irreducible Harish-Chandra modules over Kleinian singularities. There are three goals of this paper. The first is to classify anti-Poisson involutions of Kleinian singularities up to conjugation by graded Poisson automorphisms. The second is to describe the scheme-theoretic fixed point loci of Kleinian singularities under anti-Poisson involutions. The last and the main goal is to describe the scheme-theoretic preimages of the fixed point loci under minimal resolutions of Kleinian singularities, which are singular Lagrangian subvarieties in the minimal resolutions whose irreducible components are 's and 's.

Paper Structure

This paper contains 46 sections, 73 theorems, 177 equations, 20 tables.

Table of Contents

  1. Introduction
  2. Kleinian singularities
  3. Nilpotent orbits
  4. Harish-Chandra modules
  5. Main results
  6. Structure of the paper
  7. Preliminaries
  8. Reminder on Kleinian singularities
  9. Intersection pairing
  10. Poisson brackets on Kleinian singularities
  11. Anti-Poisson involutions and their fixed point loci
  12. Anti-Poisson involutions
  13. Normalizer and graded automorphisms
  14. Type $A_n$
  15. Type $D_n,\ n$ even
  16. ...and 31 more sections

Key Result

Proposition 1

There are finitely many anti-Poisson involutions on $X:=\mathbb{C}^2/\Gamma$ up to conjugation by Poisson automorphisms. They can all be written out explicitly in terms of generators of $\mathbb{C}[X]$.

Theorems & Definitions (130)

  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Proposition 4: \ref{['multofcomponent']}
  • Remark 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Definition 2.1
  • Example 2.2
  • ...and 120 more