Noncommutative projective partial resolutions and quiver varieties

2024/06/02 by Gammelgaard, Søren, Gyenge, Ádám
#14A22 #14E16 #16G20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2406.00709

Abstract

Let Γ∈ SL₂(ℂ) be a finite subgroup. We introduce a class of projective noncommutative surfaces ℙ²_I, indexed by a set of irreducible Γ-representations. Extending the action of Γ from ℂ² to ℙ², we show that these surfaces generalise both [ℙ²/Γ] and ℙ²/Γ. We prove that isomorphism classes of framed torsion-free sheaves on any ℙ²_I carry a canonical bijection to the closed points of appropriate Nakajima quiver varieties. In particular, we provide geometric interpretations for a class of Nakajima quiver varieties using noncommutative geometry. Our results partially generalise several previous results on such quiver varieties.

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