Kirillov has described a McKay correspondence for finite subgroups of PSL_{2}(C) that associates to each `height' function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on the projective line P^1 and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations.
The main goal of this paper is to develop an analogous story for the cotangent bundle of P^1. We show that each height function gives rise to a derived equivalence between equivariant sheaves on the cotangent bundle T*P^1 and modules over the preprojective algebra of an affine Dynkin quiver. These different equivalences are related by spherical twists, which take the place of the reflection functors for P^1.
We consider the problem of comparing t-structures under the derived McKay correspondence and for tilting equivalences. We relate the t-structures using certain natural torsion theories. As an application, we give a criterion for rationality for surfaces with a tilting bundle. In particular we show that every smooth projective surface which admits a full, strong, exceptional collection of line bundles is rational.
We give a new moduli construction of the minimal resolution of the singularity of type 1/r(1,a) by introducing the Special McKay quiver. To demonstrate that our construction trumps that of the G-Hilbert scheme, we show that the induced tautological line bundles freely generate the bounded derived category of coherent sheaves on X by establishing a suitable derived equivalence. This gives a moduli construction of the Special McKay correspondence for abelian subgroups of GL(2).
According to McKay (1980) the irreducible characters of finite subgroups of SU(2) are in a natural 1-1 correspondence with the extended Coxeter-Dynkin graphs of type ADE. We show that the character values themselves can be given by an uniform formula, as special values of polynomials which arise naturally as numerators of Poincare series associated to finite subgroups of SU(2) acting on polynomials in two variables. These polynomials have been the subject of a number of investigations, but their interpretation as characters has apparently not been noticed.
In this paper, using the quantum McKay correspondence, we construct the "derived category" of G-equivariant sheaves on the quantum projective line at a root of unity. More precisely, we use the representation theory of U_{q}sl(2) at root of unity to construct an analogue of the symmetric algebra and the structure sheaf. The analogue of the structure sheaf is, in fact, a complex, and moreover it is a dg-algebra. Our derived category arises via a triangulated category of G-equivariant dg-modules for this dg-algebra. We then relate this to representations of the quiver (Γ, \Om), where Γis the A,D…
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…
We propose an arithmetic McKay correspondence which relates suitably defined zeta functions of some Deligne-Mumford stacks to the zeta functions of their crepant resolutions. Some examples are discussed.
The quotient of a finite-dimensional vector space by the action of a finite subgroup of automorphisms is usually a singular variety. Under appropriate assumptions, the McKay correspondence relates the geometry of nice resolutions of singularities and the representations of the group. For the Hilbert scheme of points on the affine plane, we study how different correspondences (McKay, dual McKay and multiplicative McKay) are related to each other.
The classical McKay correspondence for finite subgroups $G$ of $\SL(2,\C)$ gives a bijection between isomorphism classes of nontrivial irreducible representations of $G$ and irreducible components of the exceptional divisor in the minimal resolution of the quotient singularity $\A^2_\C/G$. Over non algebraically closed fields $K$ there may exist representations irreducible over $K$ which split over $\bar{K}$. The same is true for irreducible components of the exceptional divisor. In this paper we show that these two phenomena are related and that there is a bijection between nontrivial irreduc…
The purpose of this paper is to show how the motivic integration methods of Kontsevich, Denef-Loeser and Looijenga can be adapted to prove the McKay-Ruan correspondence, a generalization of the McKay-Reid correspondence to orbifolds that are not necessarily global quotients.
Let $G$ be a finite subgroup of $\mbox{GL}(2)$ acting on $\mathbf{A}^2\setminus\{0\}$ freely. The $G$-orbit Hilbert scheme $G\mbox{-Hilb}(\mathbf{A}^2)$ is a minimal resolution of the quotient $\mathbf{A}^2/G$. We determine the generator sheaf of the ideal defining the universal $G$-cluster over $G\mbox{-Hilb}(\mathbf{A}^2)$, which somewhat strengthens the well-known McKay correspondence for a finite subgroup of $\mbox{SL}(2)$. We also study the quiver structure of $G\mbox{-Hilb}(\mathbf{A}^2)$ at every $G$-cluster $O_{Z_y}=O_{\mathbf{A}^2}/I_y$ in terms of a collection of sort of minimal $G$-…
We prove an equivariant version of the McKay correspondence for the elliptic genus on open varieties with a torus action. As a consequence, we will prove the equivariant DMVV formula for the Hilbert scheme of points on $\C^2$.
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.
In this note, we consider crepant resolutions of the quotient varieties of smooth quintic threefolds by Gorenstein group actions. We compute their Hodge numbers via McKay correspondence. In this way, we find some new pairs $(h^{1,1},\;h^{2,1})$ of Hodge numbers of Calabi-Yau threefolds.
The present work is devoted to the study of motivic integration on quotient singularities. We give a new proof of a form of the McKay correspondence previously proved by Batyrev. The paper contains also some general results on motivic integration on arbitrary singular spaces.
We define the singular orbifold elliptic genus and $E$-function for all normal surfaces without strictly log-canonical singularities, and prove the analogue of the McKay correspondence in this setting. Our invariants generalize the stringy invariants defined by Willem Veys for this class of singularities. We show that the ability to define these invariants is closely linked to rigidity phenomena associated to the elliptic genus.
We give a dual to the McKay correspondence, involving conjugacy classes of subgroups of SU(2). We prove a determinantal formula involving both correspondences. We pose some questions concerning a non-commutative Fourier transform.
We propose a three dimensional generalization of the geometric McKay correspondence described by Gonzales-Sprinberg and Verdier in dimension two. We work it out in detail when G is abelian and C^3/G has a single isolated singularity. More precisely, we show that the Bridgeland-King-Reid derived category equivalence induces a natural geometric correspondence between irreducible representations of G and subschemes of the exceptional set of G-Hilb (C^3). This correspondence appears to be related to Reid's recipe.
There are many generalizations of the McKay correspondence for higher dimensional Gorenstein quotient singularities and there are some applications to compute the topological invariants today. But some of the invariants are completely different from the classical invariants, in particular for non-Gorenstein cases. In this paper, we would like to discuss the McKay correspondence for 2-dimensional quotient singularities via "special" representations which gives the classical topological invariants and give a new characterization of the special representations for cyclic quotient singularities in…
We show Mckay correspondence of Betti numbers of Chen-Ruan coho- mology for omnioriented quasitoric orbifolds. In previous articles with M. Poddar [8], [9], we proved the correspondence for four dimension and six dimensions. Here we deal with the general case.
We consider the quotients $X = V/G$ of a symplectic complex vector space $V$ by a finite subgroup $G \subset Sp(V)$ which admit a smooth crepant resolution $Y \to X$. For such quotients, we prove the homological McKay correspondence conjectured by M. Reid. Namely, we construct a natural basis in the homology space $H_\cdot(Y,\Q)$ whose elements are numbered by the conjugacy classes in the finite group $G$.
Let G be a finite group of automorphisms of a nonsingular complex threefold M such that the canonical bundle omega_M is locally trivial as a G-sheaf. We prove that the Hilbert scheme Y=GHilb M parametrising G-clusters in M is a crepant resolution of X=M/G and that there is a derived equivalence (Fourier- Mukai transform) between coherent sheaves on Y and coherent G-sheaves on M. This identifies the K theory of Y with the equivariant K theory of M, and thus generalises the classical McKay correspondence. Some higher dimensional extensions are possible.
We establish a correspondence between orbifold and singular elliptic genera of a global quotient. While the former is defined in terms of the fixed point set of the action, the latter is defined in terms of the resolution of singularities. As a byproduct, the second quantization formula of Dijkgraaf, Moore, Verlinde and Verlinde is extended to arbitrary Kawamata log-terminal pairs.
We prove that the equivariant derived category for a finite subgroup of GL(3,C) has a semi-orthogonal decomposition into the derived category of a certain partial resolution, called a maximal Q-factorial terminalization, of the corresponding quotient singularity and a relative exceptional collection. This is a generalization of a result of Bridgeland, King and Reid.
This is a write-up of my talk at the Conference on algebraic structures in Montreal, July 2003. I try to give a brief informal introduction to the proof of Y. Ruan's conjecture on orbifold cohomology multiplication for symplectic quotient singularities given in V. Ginzburg and D. Kaledin, math.AG/0212279. Version 2: minor changes, added some references.
We give a conceptual proof that the Poincaré series of the coordinate algebra of a Kleinian singularity and of a Fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two Coxeter elements. These Coxeter elements are interpreted geometrically, using triangulated categories and spherical twist functors.
In this paper we prove an analogue of the McKay correspondence for Landau-Ginzburg models. Our proof is based on the ideas introduced by T. Bridgeland, A. King and M. Reid, which reformulate and generalize the McKay correspondence in the language of derived categories, along with the techniques introduced by J.-C. Chen.
We study the images of tautological bundles on Hilbert schemes of points on surfaces and their wedge powers under the derived McKay correspondence. The main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs. As an application, we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves. In particular, we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface.
In most cases where it had been shown to exist the derived McKay correspondence D(Y) --> D^G(C^n) can be written as a Fourier-Mukai transform which sends point sheaves of the crepant resolution Y to pure sheaves in D^G(C^n). We give a sufficient condition for an object of D^G(Y x C^n) to be the defining object of such a transform. We use it to construct the first example of the derived McKay correspondence for a non-projective crepant resolution of C^3/G. Along the way we extract some more geometric sense out of the Intersection Theorem and learn to explicitly compute theta-stable families of …
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.
In this paper we show that for any affine complete rational surface singularity there is a correspondence between the dual graph of the minimal resolution and the quiver of the endomorphism ring of the special CM modules. We thus call such an algebra the reconstruction algebra. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2,C), it means that the endomorphism ring of the special CM C[[x,y]]^G modules can be used to build the dual graph of the…
The boundary chiral ring of a 2d gauged linear sigma model on a Kähler manifold $X$ classifies the topological D-brane sectors and the massless open strings between them. While it is determined at small volume by simple group theory, its continuation to generic volume provides highly non-trivial information about the $D$-branes on $X$, related to the derived category $D^\flat(X)$. We use this correspondence to elaborate on an extended notion of McKay correspondence that captures more general than orbifold singularities. As an illustration, we work out this new notion of McKay correspondence fo…
We announce a result on quantum McKay correspondence for disc invariants of outer legs in toric Calabi-Yau 3-orbifolds, and illustrate our method in a special example $[\mathbb C^3 /\mathbb Z_5 (1, 1, 3)]$.
A Calabi-Yau orbifold is locally modeled on C^n/G where G is a finite subgroup of SL(n, C). In dimension n=3 a crepant resolution is given by Nakamura's G-Hilbert scheme. This crepant resolution has a description as a GIT/symplectic quotient. We use tools from global analysis to give a geometrical generalization of the McKay Correspondence to this case.
We prove the quantum McKay correspondence formulae conjectured by J. Bryan and A. Gholampour for the type D (binary) polyhedral groups in SU(2) and SO(3). We use the method of induction by the WDVV equation and from the normal subgroups by J. Bryan and A. Gholampour, and the polynomiality technique developed in this article. We are also based on the validity of the corresponding conjecture for type A groups, which is proved by T. Coates, A. Corti, H. Iritani, and H.-H. Tseng.
The classical McKay correspondence establishes an explicit link from the representation theory of a finite subgroup G of SU(2) and the geometry of the minimal resolution of the quotient of the affine plane by G. In this paper we discuss a possible generalization of the McKay correspondence to the case when G is replaced with a cocompact discrete subgroup of the universal cover of SU(1,1) such that its image in PSU(1,1) is a cocompact fuchsian group with quotient of genus 0. We establish a correspondence between a certain class of finite-dimensional unitary representations of G and vector bundl…
For a finite Abelian subgroup A of SL(3,C), Ito and Nakajima proved that the tautological bundles on the A-Hilbert scheme Y = A-Hilb(C^3) form a basis of the K-theory of Y. We establish the relations between these bundles in the Picard group of Y and hence, following a recipe introduced by Reid, construct an explicit basis of the integral cohomology of Y in one-to-one correspondence with the irreducible representations of A.
We show that the derived category of coherent sheaves on the quotient stack of the affine plane by a finite small subgroup of the general linear group is obtained from the derived category of coherent sheaves on the minimal resolution by adding a semiorthogonal summand with a full exceptional collection. The proof is based on an explicit construction in the abelian case, together with the analysis of the behavior of the derived categories of coherent sheaves under root constructions.
The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres over the origin reveal similar properties. We construct a morphism between these two resolutions, contracting exactly the excess part of the exceptional fibre. This construction is motivated by the study of some pencils of K3-surfaces arising as minimal resolutions of quotients of nodal surfaces with high symmetries.
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…
Let $G$ be a finite subgroup in SU(2), and $Q$ the corresponding affine Dynkin diagram. In this paper, we review the relation between the categories of $G$-equivariant sheaves on $P^1$ and $Rep Q_h$, where $h$ is an orientation of $Q$, constructing an explicit equivalence of corresponding derived categories.
Let $G\subset SL_2(C)\subset SL_3(C)$ be a finite group. We compute motivic Pandharipande-Thomas and Donaldson-Thomas invariants of the crepant resolution $Hilb^G(C^3)$ of $C^3/G$ generalizing results of Gholampour and Jiang who computed numerical DT/PT invariants using localization techniques. Our formulas rely on the computation of motivic Donaldson-Thomas invariants for a special class of quivers with potentials. We show that these motivic Donaldson-Thomas invariants are closely related to the polynomials counting absolutely indecomposable quiver representations over finite fields introduce…
When the quotient of a symplectic vector space by the action of a finite subgroup of symplectic automorphisms admits as a crepant projective resolution of singularities the Hilbert scheme of regular orbits of Nakamura, then there is a natural isomorphism between the Grothendieck group of this resolution and the representation ring of the group, given by the Bridgeland-King-Reid map. However, this isomorphism is not compatible with the ring structures. For the Hilbert scheme of points on the affine plane, we study the multiplicative behavior of this map.
