Symplectic resolutions of the Hilbert squares of ADE surface singularities

TL;DR

This paper investigates symplectic resolutions of Hilbert squares of surfaces with ADE singularities, characterizing these singularities via their resolutions and relating them to Slodowy slices in ADE-type nilpotent cones.

Contribution

It provides a detailed analysis of symplectic resolutions for Hilbert schemes of points on ADE surface singularities and links these to Slodowy slices in nilpotent cones.

Findings

Singularities are characterized by central fibers of symplectic resolutions.

Such singularities are isomorphic to ADE-type Slodowy slices.

The work connects symplectic geometry with Lie algebra theory.

Abstract

We study symplectic resolutions of the Hilbert scheme of two points on a surface with one ADE-singularity. We also characterize such singularities by central fibers of their symplectic resolutions. As an application, we show that these singularities are isomorphic to the Slodowy slices which are transversal to the `sub-subregular' orbits in the nilpotent cones of ADE-types.