Euler characteristics of Hilbert schemes of points on surfaces with simple singularities
Euler characteristics of Hilbert schemes of points on surfaces with simple singularities
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具有简单奇点的曲面上点的希尔伯特格式的欧拉特征
DOI：
10.1093/imrn/rnw139
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Bal'azs SzendrHoi
中科院分区：
文献类型：
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作者：
'Ad'am Gyenge;A. N'emethi;Bal'azs SzendrHoi
这是关于具有简单(Kleian)奇点的曲面上的点的Hilbert格式的欧拉特征生成系列的猜想和结果的公告。对于一个商曲面C^2/G，其中G是SL(2，C)的有限子群，我们用李论数据猜想了这个生成级数的一个公式，这与已有的A型奇点的结果是一致的。我们宣布了D型奇点猜想的一个证明。我们猜想中的生成级数可以被视为相应的(扩展的)仿射李代数的基本表示的一个特殊特征；我们讨论了这一事实的可能的表示论结果。我们的结果，分别是猜想，暗示了A型和D型分别具有任意简单奇点的曲面的母函数的模性，证实了S-对偶的预言。
This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite subgroup of SL(2, C), we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type A singularities. We announce a proof of our conjecture for singularities of type D. The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding (extended) affine Lie algebra; we discuss possible representation-theoretic consequences of this fact. Our results, respectively conjectures, imply the modularity of the generating function for surfaces with type A and type D, respectively arbitrary, simple singularities, confirming predictions of S-duality.