Birational geometry of quiver varieties and other GIT quotients

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 $V /!/{\theta} G$. If $V/!/{\theta} G$ is a crepant resolution of $Y!!:= V/!/_{0} G$, then every projective crepant resolution of $Y$ is obtained by varying $\theta$. Under suitable conditions, we show that this is the case for quiver varieties and hypertoric varieties. Similarly, for any finite subgroup $\Gamma\subset \mathrm{SL}(3,\mathbb{C})$ whose nontrivial conjugacy classes are all junior, we obtain a simple geometric proof of the fact that every projective crepant resolution of $\mathbb{C}^3/\Gamma$ is a fine moduli space of $\theta$-stable $\Gamma$-constellations.