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 upper V slash slash Subscript theta Baseline upper G V / / θ G $V{{/\!\!/\!}}_\theta G$ . If upper V slash slash Subscript theta Baseline upper G V / / θ G $V{{/\!\!/\!}}_\theta G$ is a crepant resolution of upper Y colon equals upper V slash slash Subscript 0 Baseline upper G Y := V / / 0 G $Y\!\!:= V{{/\!\!/\!}}_0 G$ , then every projective crepant resolution of Y is obtained by varying theta θ $\theta$ . Under suitable conditions, we show that this is the case for quiver varieties and hypertoric varieties. Similarly, for any finite subgroup normal upper Gamma subset of upper S upper L left parenthesis 3 comma double struck upper C right parenthesis Γ ⊂ SL ⁡ ( 3 , C ) $\Gamma\subset \operatorname{SL}(3,{{\mathbb{C}}})$ </jats:inline