Combinatorial Reid's recipe for consistent dimer models

Reid's recipe for a finite abelian subgroup $G\subset \text{SL}(3,\mathbb{C})$ is a combinatorial procedure that marks the toric fan of the $G$-Hilbert scheme with irreducible representations of $G$. The geometric McKay correspondence conjecture of Cautis--Logvinenko that describes certain objects in the derived category of $G\text{-Hilb}$ in terms of Reid's recipe was later proved by Logvinenko et al. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence of Bocklandt--Craw--Quintero-V\'{e}lez. Our main tool generalises the jigsaw transformations of Nakamura to consistent dimer models. Comment: 29 pages, published version

Elliptic classes, McKay correspondence and theta identities

We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to the equivariant local situation. We study theta function identities having a geometric origin. In the case of quotient singularities $${\mathbb {C}}^n/G$$ 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 $$A_n$$ A n leads to a remarkable formula.

From the Trinity ( A 3 , B 3 , H 3 ) to an ADE correspondence

In this paper, we present novel ADE correspondences by combining an earlier induction theorem of ours with one of Arnold's observations concerning Trinities, and the McKay correspondence. We first extend Arnold's indirect link between the Trinity of symmetries of the Platonic solids ( A 3 , B 3 , H 3 ) and the Trinity of exceptional four-dimensional root systems ( D 4 , F 4 , H 4 ) to an explicit Clifford algebraic construction linking the two ADE sets of root systems ( I 2 ( n ), A 1 × I 2 ( n ), A 3 , B 3 , H 3 ) and ( I 2 ( n ), I 2 ( n )× I 2 ( n ), D 4 , F 4 , H 4 ). The latter are connected through the McKay correspondence with the ADE Lie algebras ( A n , D n , E 6 , E 7 , E 8 ). We show that there are also novel indirect as well as direct connections between these ADE root systems and the new ADE set of root systems ( I 2 ( n ), A 1 × I 2 ( n ), A 3 , B 3 , H 3 ), resulting in a web of three-way ADE correspondences between three ADE sets of root systems.

On a logarithmic version of the derived McKay correspondence

We globalize the derived version of the McKay correspondence of Bridgeland, King and Reid, proven by Kawamata in the case of abelian quotient singularities, to certain logarithmic 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 logarithmic blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under logarithmic blow-up, up to Morita equivalence.

Extended McKay correspondence for quotient surface singularities

Let G be a finite subgroup of GL(2) acting on A2/{0} freely. The G-orbit Hilbert scheme G-Hilb(A2) is a minimal resolution of the quotient A2/G as given by A. Ishii, On the McKay correspondence for a finite small subgroup of GL(2,C), J. Reine Angew. Math. 549 (2002), 221–233. We determine the generator sheaf of the ideal defining the universal G-cluster over G-Hilb(A2), which somewhat strengthens a version [10] of the well-known McKay correspondence for a finite subgroup of SL(2).