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Birational geometry of quiver varieties and other GIT quotients

Gwyn Bellamy, Alastair Craw, Travis Schedler

TL;DR

This work provides a unified, geometry-driven framework for understanding the birational maps among GIT quotients and their resolutions. By establishing Condition GIT, the authors show that a large class of quotients are relative Mori Dream Spaces and that their movable cones are controlled by the GIT chamber structure via the linearisation map. They apply this to Nakajima quiver varieties, hypertoric varieties, and certain threefold quotient singularities, proving that crepant resolutions are themselves moduli-type spaces and describing how all such resolutions arise from variation of GIT quotients. This approach yields explicit descriptions of Mov(X/Y), the Namikawa–Weyl group action, and hyperplane arrangements, circumventing reliance on the BCHM machinery and offering constructive, combinatorial tools for birational classification in this setting.

Abstract

We prove that all projective crepant resolutions of Nakajima quiver varieties satisfying natural conditions are also Nakajima quiver varieties. More generally, we classify the small birational models of many Geometric Invariant Theory (GIT) quotients by introducing a sufficient condition for the GIT quotient of an affine variety $V$ by the action of a reductive group $G$ to be a relative Mori Dream Space. Two surprising examples illustrate that our new condition is optimal. When the condition holds, we show that the linearisation map identifies a region of the GIT fan with the Mori chamber decomposition of the relative movable cone of $V /\!/_θ G$. If $V/\!/_θ G$ is a crepant resolution of $Y\!\!:= V/\!/_{0} G$, then every projective crepant resolution of $Y$ is obtained by varying $θ$. Under suitable conditions, we show that this is the case for quiver varieties and hypertoric varieties. Similarly, for any finite subgroup $Γ\subset \mathrm{SL}(3,\mathbb{C})$ whose nontrivial conjugacy classes are all junior, we obtain a simple geometric proof of the fact that every projective crepant resolution of $\mathbb{C}^3/Γ$ is a fine moduli space of $θ$-stable $Γ$-constellations.

Birational geometry of quiver varieties and other GIT quotients

TL;DR

This work provides a unified, geometry-driven framework for understanding the birational maps among GIT quotients and their resolutions. By establishing Condition GIT, the authors show that a large class of quotients are relative Mori Dream Spaces and that their movable cones are controlled by the GIT chamber structure via the linearisation map. They apply this to Nakajima quiver varieties, hypertoric varieties, and certain threefold quotient singularities, proving that crepant resolutions are themselves moduli-type spaces and describing how all such resolutions arise from variation of GIT quotients. This approach yields explicit descriptions of Mov(X/Y), the Namikawa–Weyl group action, and hyperplane arrangements, circumventing reliance on the BCHM machinery and offering constructive, combinatorial tools for birational classification in this setting.

Abstract

We prove that all projective crepant resolutions of Nakajima quiver varieties satisfying natural conditions are also Nakajima quiver varieties. More generally, we classify the small birational models of many Geometric Invariant Theory (GIT) quotients by introducing a sufficient condition for the GIT quotient of an affine variety by the action of a reductive group to be a relative Mori Dream Space. Two surprising examples illustrate that our new condition is optimal. When the condition holds, we show that the linearisation map identifies a region of the GIT fan with the Mori chamber decomposition of the relative movable cone of . If is a crepant resolution of , then every projective crepant resolution of is obtained by varying . Under suitable conditions, we show that this is the case for quiver varieties and hypertoric varieties. Similarly, for any finite subgroup whose nontrivial conjugacy classes are all junior, we obtain a simple geometric proof of the fact that every projective crepant resolution of is a fine moduli space of -stable -constellations.
Paper Structure (32 sections, 42 theorems, 42 equations, 2 figures)

This paper contains 32 sections, 42 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main result for GIT quotients
  3. Application to quiver varieties
  4. Hypertoric varieties
  5. Application to threefold quotient singularities
  6. Optimality of Condition \ref{['cond:GIT']}
  7. Background
  8. Birational geometry
  9. GIT quotients and the linearisation map
  10. Variation of GIT quotient
  11. Reconstructing relative Mori Dream Spaces by GIT
  12. GIT regions
  13. The new GIT condition and two key examples
  14. The main result
  15. Relative Mori Dream Spaces
  16. ...and 17 more sections

Key Result

Theorem 1.1

For the action of a reductive group $G$ on an affine variety $V$, suppose that a GIT chamber $C$ in $G^\vee_\mathbb{Q}$ satisfies Condition cond:GIT. For $\theta\in C$, write $X_{\theta{}}:=V/\!\!/\!_\theta \, G$. In particular, the $\mathbb{Q}$-factorial small birational models of $X_{\theta{}}$ are the birational models $V/\!\!/\!_\zeta \, G$ determined by the line bundles $L_C(\zeta)$ for gene

Figures (2)

  • Figure 1: (a) slice of the fan defining $X$; (b) flops linking crepant resolutions
  • Figure 2: (a) quiver of sections $Q$ on $Y$; (b) graph indicating chambers that lie adjacent

Theorems & Definitions (104)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • ...and 94 more