Samuel Lewis

Quiver varieties in dimension four

Nakajima quiver varieties form an important class of examples of conical symplectic singularities, such as the well-studied Kleinian singularities in dimension two. In this talk I will describe a classification of the four-dimensional quiver varieties, obtained combinatorially and comprising dimension vectors in three “Types”. The theory of root systems is then applied to describe the geometric structure of these varieties in detail, such as symplectic leaves, their minimal degenerations, and the Namikawa Weyl group. If time permits, I will also discuss how one may compute the number of projective crepant resolutions admitted by the quiver varieties, and some conjectures on the resulting numerology. This is all based on joint work with Pavel Shlykov (arXiv:2510.15160).