Counting resolutions of symplectic quotient singularities
Counting resolutions of symplectic quotient singularities
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DOI：
10.1112/s0010437x15007630
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发表时间：
2014-05
影响因子：
1.8
通讯作者：
G. Bellamy
中科院分区：
文献类型：
--
作者：
G. Bellamy
设${\rm\Gamma}$是$\text{Sp}（V）$的有限子群。在这篇文章中，我们计算了商奇点$V/{\rm\Gamma}$所允许的辛分解的数量。我们的方法是比较辛商奇异的普适泊松变形与Calogero-Moser空间给出的变形。通过这种方式，我们给出了一个简单的公式的数量$\mathbb{Q}$-阶乘终结承认的辛商奇性的维数的某个Orlik-Solomon代数自然相关的Calogero-Moser变形。这个维度是明确计算的所有群体${\rm\Gamma}$，它是已知的$V/{\rm\Gamma}$承认一个辛决议。作为我们的结果的后果，我们证实了金兹伯格和Kaledin的猜想。
Let ${\rm\Gamma}$ be a finite subgroup of $\text{Sp}(V)$. In this article we count the number of symplectic resolutions admitted by the quotient singularity $V/{\rm\Gamma}$. Our approach is to compare the universal Poisson deformation of the symplectic quotient singularity with the deformation given by the Calogero–Moser space. In this way, we give a simple formula for the number of $\mathbb{Q}$-factorial terminalizations admitted by the symplectic quotient singularity in terms of the dimension of a certain Orlik–Solomon algebra naturally associated to the Calogero–Moser deformation. This dimension is explicitly calculated for all groups ${\rm\Gamma}$ for which it is known that $V/{\rm\Gamma}$ admits a symplectic resolution. As a consequence of our results, we confirm a conjecture of Ginzburg and Kaledin.