A new family of isolated symplectic singularities with trivial local fundamental group

TL;DR

This paper introduces an infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, constructed through three different geometric methods, addressing a question posed by Beauville in 2000.

Contribution

It provides the first explicit examples of such singularities, constructed via quotient, Calogero-Moser spaces, and Slodowy slices, expanding understanding of symplectic singularities.

Findings

Constructed an infinite family of symplectic singularities with trivial local fundamental group.

Presented three distinct geometric constructions for these singularities.

Answered a longstanding question of Beauville from 2000.

Abstract

We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of $\mathbb{C}^4$ by the dihedral group of order $2d$, (2) as singular points of Calogero-Moser spaces associated with dihedral groups of order $2d$ at equal parameters, (3) as singularities of a certain Slodowy slice in the $d$-fold cover of the nilpotent cone in ${\mathfrak{sl}}_d$.