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Singularities in Calogero--Moser Varieties

Gwyn Bellamy, Ruslan Maksimau, Travis Schedler

TL;DR

This work classifies singularities of Calogero–Moser varieties for wreath-product symmetry groups by parameterizing symplectic leaves, determining closure relations, and constructing transverse slices. It exploits the identification between CM varieties and framed Nakajima quiver varieties to transport the problem into the well-developed quiver-variety framework, applying Crawley–Boevey’s stratification and étale normal forms to describe leaf closures and their normalizations. The authors prove that the normalization of the closure of each symplectic leaf is itself a Calogero–Moser variety for an associated subquotient, thereby confirming a Bonnafé-type conjecture in this setting and connecting the local geometry to global CM data. The analysis covers non-zero and zero level parameters, the cyclic (type A) case with explicit ℓ-core combinatorics, and the hyperoctahedral family, with a unified approach that extends to arbitrary quiver varieties. The results provide a detailed geometric and combinatorial picture of CM singularities, with potential implications for representation theory of spherical symplectic reflection algebras and related Harish-Chandra theory.

Abstract

In this article we describe completely the singularities appearing in Calogero--Moser varieties associated (at any parameter) to the wreath product symplectic reflection groups. We do so by parameterizing the symplectic leaves in the variety, describing combinatorially the resulting closure relation and computing a transverse slice to each leaf. We also show that the normalization of the closure of each symplectic leaf is isomorphic to a Calogero--Moser variety for an associated (explicit) subquotient of the symplectic reflection group. This confirms a conjecture of Bonnafé for these groups. We use the fact that the Calogero--Moser varieties associated to wreath products can be identified with certain Nakajima quiver varieties. In particular, our result identifying the normalization of the closure of each symplectic leaf with another quiver variety holds for arbitrary quiver varieties.

Singularities in Calogero--Moser Varieties

TL;DR

This work classifies singularities of Calogero–Moser varieties for wreath-product symmetry groups by parameterizing symplectic leaves, determining closure relations, and constructing transverse slices. It exploits the identification between CM varieties and framed Nakajima quiver varieties to transport the problem into the well-developed quiver-variety framework, applying Crawley–Boevey’s stratification and étale normal forms to describe leaf closures and their normalizations. The authors prove that the normalization of the closure of each symplectic leaf is itself a Calogero–Moser variety for an associated subquotient, thereby confirming a Bonnafé-type conjecture in this setting and connecting the local geometry to global CM data. The analysis covers non-zero and zero level parameters, the cyclic (type A) case with explicit ℓ-core combinatorics, and the hyperoctahedral family, with a unified approach that extends to arbitrary quiver varieties. The results provide a detailed geometric and combinatorial picture of CM singularities, with potential implications for representation theory of spherical symplectic reflection algebras and related Harish-Chandra theory.

Abstract

In this article we describe completely the singularities appearing in Calogero--Moser varieties associated (at any parameter) to the wreath product symplectic reflection groups. We do so by parameterizing the symplectic leaves in the variety, describing combinatorially the resulting closure relation and computing a transverse slice to each leaf. We also show that the normalization of the closure of each symplectic leaf is isomorphic to a Calogero--Moser variety for an associated (explicit) subquotient of the symplectic reflection group. This confirms a conjecture of Bonnafé for these groups. We use the fact that the Calogero--Moser varieties associated to wreath products can be identified with certain Nakajima quiver varieties. In particular, our result identifying the normalization of the closure of each symplectic leaf with another quiver variety holds for arbitrary quiver varieties.

Paper Structure

This paper contains 47 sections, 73 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Symplectic leaves
  3. Transverse slices
  4. The cyclic group
  5. The hyperoctahedral group
  6. Method of proof
  7. Leaf closures in quiver varieties
  8. Outline of the article
  9. Quiver varieties
  10. Notation
  11. Graphs
  12. Root systems
  13. Quiver varieties
  14. Deframing
  15. The symplectic leaves
  16. ...and 32 more sections

Key Result

Theorem 1.1

If the leaf ${\mathfrak L}$ is labeled by the class $(\mathrm{P})$ then there exists $w \in W^{\mathrm{aff}}$ such that where $\widetilde{{\mathfrak L}}$ is the normalization of the closure of ${\mathfrak L}$ in ${\boldsymbol{\mathcal{Z}}}_{c}(\Gamma_n)$.

Theorems & Definitions (157)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 147 more