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.
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 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 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.
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 $\C^2$.
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 revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead of using cohomology, we deal with the Chow motives. More precisely, we prove that for any smooth proper two-dimensional orbifold with projective coarse moduli space, there is an isomorphism of algebra objects, in the category of complex Chow motives, between the motive of the minimal resolution and the orbifold motive. In particular, the complex Chow ring (r…
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 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 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.
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.
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.
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.
We study the behavior of a dimer model under the operation of removing a corner from the lattice polygon and taking the convex hull of the rest. This refines an operation of Gulotta, and the special McKay correspondence plays an essential role in this refinement. As a corollary, we show that for any lattice polygon, there is a dimer model such that the derived category of finitely-generated modules over the path algebra of the corresponding quiver with relations is equivalent to the derived category of coherent sheaves on a toric Calabi-Yau 3-fold determined by the lattice polygon. Our proof i…
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 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 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).
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 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…
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.
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.
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.
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…
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 $[\mathbb C^3 /\mathbb Z_5 (1, 1, 3)]$.
Canonical orders, introduced in the minimal model program for orders, are simultaneous generalisations of Kleinian singularities and their associated skew group rings. In this paper, we construct minimal resolutions of canonical orders via non-commutative cyclic covers and skew group rings. This allows us to exhibit a derived equivalence between minimal resolutions of canonical orders and the skew group ring form of the canonical order in all but one case. The Fourier-Mukai transform used to construct this equivalence allows us to make explicit, the numerical version of the McKay correspondenc…
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 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 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…
This paper addresses a long standing open problem due to Lehmer in which the triple 2,3,7 plays a notable role. Lehmer's problem asks whether there is a gap between 1 and the next smallest algebraic integer with respect to Mahler measure. The question has been studied in a wide range of contexts including number theory, ergodic theory, hyperbolic geometry, and knot theory; and relates to basic questions such as describing the distribution of heights of algebraic integers, and of lengths of geodesics on arithmetic surfaces. This paper focuses on the role of Coxeter systems in Lehmer's problem. …
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.
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…
