Related papers: Geometrical McKay Correspondence for Isolated Sing…

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity C^3/G. The classical McKay correspondence describes the classical…

Let X/G be a 3-dimensional Calabi-Yau orbifold with codimension 2 singularities. The topology of crepant resolutions of X/G is described by the McKay correspondence (Reid, Ito). We study Calabi-Yau 3-folds Y that arise by deforming the…

We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry. We give a universal construction of Calabi-Yau algebras in terms of a noncommutative symplectic DG algebra resolution.…

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…

We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Calabi-Yau orbifolds by viewing the open theories as sections of Givental's symplectic vector space and the correspondence as a linear map of…

We present, in explicit matrix representation and a modernity befitting the community, the classification of the finite discrete subgroups of G_2 and compute the McKay quivers arising therefrom. Of physical interest are the classes of N=1…

We present a local computation of deformations of the tangent bundle for a resolved orbifold singularity C^d/G. These correspond to (0,2)-deformations of (2,2)-theories. A McKay-like correspondence is found predicting the dimension of the…

We study first order deformations of the tangent sheaf of resolutions of Calabi-Yau threefolds that are of the form $\mathbb{C}^3/Z_r$, focusing on the cases where the orbifold has an isolated singularity. We prove a lower bound on the…

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.…

This is the final draft, containing very minor proof-reading corrections. Let G in SL(n,\C) be a finite subgroup and \fie: Y -> X = \C^n/G any resolution of singularities of the quotient space. We prove that crepant exceptional prime…

Let M be a quasiprojective algebraic manifold with K_M=0 and G a finite automorphism group of M acting trivially on the canonical class K_M; for example, a subgroup G of SL(n,C) acting on C^n in the obvious way. We aim to study the quotient…

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…

This is a rough write-up of my lecture at Kinosaki and two lectures at RIMS workshops in Dec 1996, on work in progress that has not yet reached any really worthwhile conclusion, but contains lots of fun calculations. History of Vafa's…

Let $G$ be a nontrivial finite subgroup of $\SL_n(\C)$. Suppose that the quotient singularity $\C^n/G$ has a crepant resolution $\pi\colon X\to \C^n/G$ (i.e. $K_X = \shfO_X$). There is a slightly imprecise conjecture, called the McKay…

The three-dimensional McKay correspondence seeks to relate the geometry of crepant resolutions of Gorenstein $3$-fold quotient singularities $\mathbb{A}^3/G$ with the representation theory of the group $G$. The first crepant resolution…

Let G be a finite subgroup of SL(n,C). If a quotient variety C^n/G has a crepant resolution, then its Euler number equals to the number of conjugacy classes of G, which is a weak version of the McKay correspondence. In this paper, we…

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…

We describe the mirror of the Z orbifold as a representation of a class of generalized Calabi-Yau manifolds that can be realized as manifolds of dimension five and seven. Despite their dimension these correspond to superconformal theories…

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…

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…