Punctual Hilbert schemes for Kleinian singularities as quiver varieties

2019/10/29 by Craw, Alastair, Gammelgaard, Søren, Gyenge, Ádám +1 · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1910.13420

Abstract

For a finite subgroup Γ⊂ SL(2,ℂ) and n≥ 1, we construct the (reduced scheme underlying the) Hilbert scheme of n points on the Kleinian singularity ℂ²/Γ as a Nakajima quiver variety for the framed McKay quiver of Γ, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of stability parameter.

Cited by

• Some Semi-Classical Noncommutative Resolutions of Kleinian Singularities
• From Hitchin Systems to Rational Elliptic Surfaces with C*-actions via Orbifold Hilbert Schemes

Related

• Quiver Varieties and Framed Sheaves on Compactified Kleinian Singularities
• Hilbert schemes for crepant partial resolutions
• The fixed point locus of the smooth Jordan quiver variety under the action of the finite subgroups of SL₂(ℂ)
• Birational geometry of symplectic quotient singularities
• An introduction to Hilbert schemes of points on ADE singularities