论文标题
箭袋品种和其他git商的生育几何形状
Birational geometry of quiver varieties and other GIT quotients
论文作者
论文摘要
我们证明，满足自然条件的Nakajima Quiver品种的所有投射毛茸茸的分辨率也是Nakajima Quiver品种。更普遍地，我们通过通过还原群体$ g $的作用引入了仿射品种$ v $的git商来对许多几何不变理论（GIT）商的小型模型进行分类。两个令人惊讶的例子说明我们的新状况是最佳的。当条件成立时，我们表明线性化图可以通过$ v /\！ /_θG$的相对可移动锥的MORI腔室分解来识别GIT风扇的区域。如果$ v/\！/_θG$是$ y \！\！\！：= v/\！/_ {0} g $的可培养物分辨率，则每个投射的$ y $的射击分辨率都是通过不同的$θ$获得的。在适当的条件下，我们表明箭量品种和高血压品种就是这种情况。同样，对于任何有限的子组$γ\ subset \ mathrm {slrm {sl}（3，\ mathbb {c}）$，其非平凡的共轭类都是初级的，我们可以获得一个简单的几何证据，证明了每个投射的jathbb {c}^3/γ$ fine moduli的投影式crepants $ space的事实$γ$ - 构造。
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.
