Singularities in Calogero--Moser Varieties

TL;DR

This paper thoroughly characterizes the singularities in Calogero--Moser varieties linked to wreath product symplectic reflection groups, confirming a conjecture and connecting these varieties to Nakajima quiver varieties.

Contribution

It provides a complete description of singularities, parametrizes symplectic leaves, and proves the normalization of leaf closures is isomorphic to related Calogero--Moser varieties, confirming Bonnafé's conjecture.

Findings

Complete description of singularities in Calogero--Moser varieties

Normalization of leaf closures is isomorphic to Calogero--Moser varieties for subquotients

Identification of these varieties with Nakajima quiver varieties

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 Bonnaf\'e 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.