Various algebraic structures have recently appeared in a parallel way in the framework of Hilbert schemes of points on a surface and respectively in the framework of equivariant K-theory [N1,Gr,S2,W], but direct connections are yet to be clarified to explain such a coincidence. We provide several non-trivial steps toward establishing our main conjecture on the isomorphism between the Hilbert quotient of the affine space $\C^{2n}$ by the wreath product $\G ~ S_n$ and Hilbert schemes of points on the minimal resolution of a simple singularity $\C^2 /\G$. We discuss further various implications of our main conjecture. We obtain a key ingredient toward a direct isomorphism between two forms of McKay correspondence in terms of Hilbert schemes [N1, Gr, N2] and respectively of wreath products [FJW]. We in addition establish a direct identification of various algebraic structures appearing in two different setups of equivariant K-theory [S2, W].
Wreath Macdonald polynomials and categorical McKay correspondence
Mark Haiman将Macdonald正性猜想简化为关于平面上点的Hilbert方案的几何的陈述，并将对称群替换为圈积$S_n\ltimes (Z/r Z)^n$来表述该猜想的推广。他已经证明了原来的猜想，建立几何声明的希尔伯特计划，作为副产品，他得到了一个推导等价相干层之间的希尔伯特计划和相干层的orbifold商的${\mathbb A}^{2n}$的对称群$S_n$。第一作者和Kaledin通过正特征量子化得到了任何辛商奇点的类似导出等价的一个简短证明。在本说明中，我们显示的性质导出等价的圈积的广义Macdonald正性。
The condensation of closed string tachyons localized at the fixed point of a C^d/Γorbifold can be studied in the framework of renormalization group flow in a gauged linear sigma model. The evolution of the Higgs branch along the flow describes a resolution of singularities via the process of tachyon condensation. The study of the fate of D-branes in this process has lead to a notion of a ``quantum McKay correspondence.'' This is a hypothetical correspondence between fractional branes in an orbifold singularity in the ultraviolet with the Coulomb and Higgs branch branes in the infrared. In this…
Derived McKay correspondence via pure-sheaf transforms
在大多数情况下，当它被证明存在时，导出的McKay对应D（Y）-> D^G（C^n）可以被写为傅里叶-向井变换，该变换将具有crepant分辨率Y的点层发送到D^G（C^n）中的纯层。我们给出了D^G（Y x C^n）的对象是这种变换的定义对象的一个充分条件。我们用它来构造C^3/G的非投射收缩分解的McKay对应的第一个例子。在此过程中，我们从相交定理中提取更多的几何意义，并学习显式计算G星座的θ稳定族及其直接变换。
We globalize the derived version of the McKay correspondence of Bridgeland-King-Reid, proven by Kawamata in the case of abelian quotient singularities, to certain log algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible log blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under log blow-up, up to Morita equivalence.
We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to equivariant local situation. We study theta function identities having geometric origin. In the case of quotient singularities $\mathbb C^n/G$, where $G$ is a finite group the theta identities arise from McKay correspondence. The symplectic singularities are of special interest. The Du Val surface singularity $A_n$ leads to a remarkable formula.
The \(v\)-function appears in the wild McKay correspondence and it is an invariant of a Galois extension over a local field which measures about the ramification of extension. The ramification filtration is also such invariant. Yasuda presented the problem whether the \(v\)-function is determined by ramification filtration. In this paper, we give a formula for the \(v\)-function for 2-dimensional representations of the bicyclic group of order \(p^2\) with \(p\) the characteristic of our base field. Using this formula, we construct an example of \(v\)-function not determined by the ramification…
A finite subgroup of ${\rm SL}_2(\CC)$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups $G$ of ${\rm SL}_3(\CC)$. They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that w…
