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All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

Gwyn Bellamy, Alastair Craw, Steven Rayan, Travis Schedler, Hartmut Weiss

TL;DR

We establish a Poisson isomorphism between the exceptional four-dimensional quotient $C^4/G$ with $G=Q_8\times_{Z_2}D_8$ and the affine hyperpolygon space $X_5(0)$, providing a uniform quiver-variety framework that realizes all $81$ projective crepant resolutions of $C^4/G$ as hyperpolygon spaces $X_5(\\theta)$. By exploiting the star-shaped quiver description, we completely describe the birational geometry of $X_n(\\theta)$, including the movable cone and Namikawa Weyl group action, and show that the number of resolutions equals the number of chambers in a fixed chamber in stability space, yielding $81$ for $n=5$ and $1684$ for $n=6$. We prove that for $n>5$ the affine hyperpolygon spaces $X_n(0)$ are not finite quotient singularities, situating the $n=4$–$5$ cases within the broader Hamiltonian-reduction/quiver-variety paradigm and highlighting why this phenomenon does not extend to higher dimensions. The work also analyzes the hyperkähler geometry, proving the $n=5$ case should be QALE in the generic metric class, while the higher-$n$ geometries fail to be QALE, and discusses connections to the McKay correspondence in this higher-rank setting.

Abstract

We demonstrate that the linear quotient singularity for the exceptional subgroup G in Sp(4,C) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of the quotient singularity C4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n-pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2n - 6; for example, we show that there are 1684 projective crepant resolutions when n = 6. We also prove that the resulting affine cones are not quotient singularities for n >= 6.

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

TL;DR

We establish a Poisson isomorphism between the exceptional four-dimensional quotient with and the affine hyperpolygon space , providing a uniform quiver-variety framework that realizes all projective crepant resolutions of as hyperpolygon spaces . By exploiting the star-shaped quiver description, we completely describe the birational geometry of , including the movable cone and Namikawa Weyl group action, and show that the number of resolutions equals the number of chambers in a fixed chamber in stability space, yielding for and for . We prove that for the affine hyperpolygon spaces are not finite quotient singularities, situating the – cases within the broader Hamiltonian-reduction/quiver-variety paradigm and highlighting why this phenomenon does not extend to higher dimensions. The work also analyzes the hyperkähler geometry, proving the case should be QALE in the generic metric class, while the higher- geometries fail to be QALE, and discusses connections to the McKay correspondence in this higher-rank setting.

Abstract

We demonstrate that the linear quotient singularity for the exceptional subgroup G in Sp(4,C) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of the quotient singularity C4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n-pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2n - 6; for example, we show that there are 1684 projective crepant resolutions when n = 6. We also prove that the resulting affine cones are not quotient singularities for n >= 6.
Paper Structure (29 sections, 32 theorems, 29 equations, 1 figure)

This paper contains 29 sections, 32 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. An exceptional quotient singularity
  3. A hyperplane arrangement
  4. The birational geometry of hyperpolygon spaces
  5. Beyond the movable cone
  6. Quotient singularities and Hamiltonian reduction.
  7. The hyperkähler geometry of hyperpolygon spaces
  8. Hyperpolygon spaces
  9. Hyperpolygon spaces as quiver varieties
  10. Combinatorics of star shaped root systems
  11. Symplectic leaves
  12. The hyperplane arrangement
  13. The semi-invariant ring
  14. All 81 symplectic resolutions are quiver varieties
  15. The birational geometry of hyperpolygon spaces
  16. ...and 14 more sections

Key Result

Theorem 1.1

There is a Poisson isomorphism $\mathbb{C}^4 / G {\;\stackrel{_\sim}{\longrightarrow}\;} X_5(0):= \mathfrak{M}_{0}(\boldsymbol{v},0)$.

Figures (1)

  • Figure 1: The quiver $Q$ (left) and the doubled quiver $\overline{Q}$ (right), each with $n+1$ vertices.

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 60 more