Nakajima quiver varieties in dimension four

Samuel Lewis (University of Oxford)

Wednesday 28th October 16:00-17:00
Maths 311B

Abstract

Nakajima quiver varieties form an important class of examples of conical symplectic singularities. Notable examples of these include Hilbert schemes of points on surfaces, nilpotent orbit closures, and Kleinian singularities. In this talk I will describe a classification, obtained combinatorially, of the four-dimensional quiver varieties. The theory of root systems is then applied to describe the geometric structure of these varieties, obtaining a classification of their symplectic leaves, the minimal degenerations between them, and their Namikawa Weyl group. If time permits, I will also discuss how one may compute the number of projective crepant resolutions admitted by a quiver variety, and ongoing work in this direction. This is all based on joint work (arXiv:2510.15160) with Pavel Shlykov.

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