Nakajima quiver varieties in dimension four

TL;DR

The authors provide a complete combinatorial classification of 4d Nakajima quiver varieties by identifying all α in Upsigma0 with $p(oldsymbol{eta})=2$, detailing the resulting symplectic leaves and minimal degenerations, and computing Namikawa Weyl groups. They show that each admissible dimension vector yields a unique (up to isomorphism) conical symplectic singularity, and they express varieties as products or symmetric powers of Kleinian singularities in most cases. The work yields a full description of isotropic decompositions, resolves the 4d (2,2) case, and demonstrates that the exceptional group quotient $G_4$ cannot be realized as a quiver variety, addressing a question in the field. By leveraging secondary and Coxeter arrangements, they count projective symplectic resolutions and relate these counts to Weyl-group data, providing new invariants to distinguish 4d quiver varieties. The results advance the understanding of 4d conical symplectic singularities and connect to broader themes like symplectic duality and deformation quantizations, with computational methods supporting explicit enumeration.

Abstract

This paper classifies all 4d Nakajima quiver varieties through a combinatorial approach. For each such variety, we describe the symplectic leaves and minimal degenerations between them. Using the resulting Hasse diagrams and secondary hyperplane arrangements, we fully classify the quiver varieties up to isomorphism, a step in the problem of classifying all 4d conical symplectic singularities and the (2, 2) case of quiver varieties. As an application, we answer in the negative a question posed by Bellamy, Craw, Rayan, Schedler, and Weiss regarding whether the $G_4$ quotient singularity (or its projective crepant resolutions) can be realised as a quiver variety.

Theorems & Definitions (81)

• Theorem 1
• Remark 2
• Theorem 3
• Remark 4
• Theorem 5
• Corollary 6
• Remark 9
• Remark 11
• Definition 12
• Lemma 13