Related papers: The McKay correspondence in type $D_4$ via VGIT

The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a K3 surface with a non symplectic action of Z_4. In this paper we study the geometry of the one dimensional family of K3 surfaces…

In this article, we revisit the classical McKay correspondence via homological mirror symmetry. Specifically, we demonstrate how this correspondence can be articulated as a derived equivalence between the category of vanishing cycles…

We provide a construction of 81 symplectic resolutions of a 4-dimensional quotient singularity obtained by an action of a group of order 32. The existence of such resolutions is known by a result of Bellamy and Schedler. Our explicit…

In this paper the relationship between the classical description of the resolution of quotient singularities and the string picture is reviewed in the context of N=(2,2) superconformal field theories. A method for the analysis of quotients…

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…

We compute the cylindrical contact homology of the links of the simple singularities. These manifolds are contactomorphic to $S^3/G$ for finite subgroups $G\subset SU(2)$. We perturb the degenerate contact form on $S^3/G$ with a Morse…

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 describe the GIT compactification for the moduli space of smooth quintic surfaces in projective space. In particular, we show that a normal quintic surface with at worst an isolated double point or a minimal elliptic singularity is…

Let $V$ be a complex vector space on which a finite group $G$ acts by linear transformations. Let $W = V \oplus V^*$ be the sum of $V$ with its dual $V^*$. We prove that if the quotient $W/G$ admits a smooth crepant resolution, then the…

Mukai showed that the GIT quotient $\operatorname{Gr}(7,16) /\!/ \operatorname{Spin}(10)$ is a birational model of the moduli space of Deligne-Mumford stable genus 7 curves $\overline{M}_7$. The key observation is that a general smooth…

We describe the GIT compactification of the moduli space of cubic fourfolds, with a special emphasis on the role played by singularities. Our main result is that a cubic fourfold with only isolated simple (A-D-E) singularities is GIT…

We announce an explicit description of the strictly semistable boundary of the GIT moduli space of quintic threefolds. For the natural action of \(\mathrm{SL}(5)\) on \(\mathbb P(\mathrm{Sym}^5\mathbb C^5)\), we classify the 38 boundary…

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…

The purpose of this paper is to construct a crepant resolution of quotient singularities by trihedral groups ( finite subgroups of SL(3,C) of certain type ), and prove that each Euler number of the minimal model is equal to the number of…

M-theory geometric engineering on non-compact Calabi-Yau fourfolds (CY4) produces 3d theories with 4 supercharges. Carefully establishing a dictionary between the geometry of the CY4 and the QFT in the transverse directions remains, to a…

In this note, we describe a a systematic procedure to find toric crepant resolutions of orbifold vertex, and show that the generating series of certain disc invariants of the orbifold vertex can be suitably identified with the generating…

The derived McKay correspondence conjecture says that there is an equivalence of triangulated categories between the bounded derived categories of commutative and non-commutative crepant resolutions of a Gorenstein singularity. We will…

In this work we develop a simple protocol to construct interior solutions through Gravitational Decoupling by the Minimal Geometric Deformation extended satisfying the vanishing complexity condition. The method is illustrated by using…

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…

We construct a form of the $D_4^+$-singularity of fronts in $R^3$ which uses coordinate transformation on the source and isometry on the target. As an application, we calculate differential geometric invariants near the $D_4^+$-singularity,…