On the existence of crepant resolutions of Gorenstein abelian quotient singularities in dimensions ≥4

Abstract

For which finite subgroups G of SL(r, C), r ≥ 4, are there crepant desingularizations of the quotient space C r /G? A complete answer to this question (also known as "Existence Problem" for such desingularizations) would classify all those groups for which the high-dimensional versions of McKay correspondence are valid. In the paper we consider this question in the case of abelian finite subgroups of SL(r, C) by using techniques from toric and discrete geometry. We give two necessary existence conditions, involving the Hilbert basis elements of the cone supporting the junior simplex, and an Upper Bound Theorem, respectively. Moreover, to the known series of Gorenstein abelian quotient singularities admitting projective, crepant resolutions (which are briefly recapitulated) we add a new series of non-c.i. cyclic quotient singularities having this property.

Branching law for finite subgroups of 𝐒𝐋3ℂ$\mathbf {SL}_3\mathbb {C}$ and McKay correspondence

“…Let h be a Cartan subalgebra of sl 3 C and let $ 1 ; $ 2 be the corresponding fundamental weights. For .m; n/ 2 N 2 , the restriction m;n j of the irreducible representation m;n of highest weight m$ 1 C n$ 2 of SL 3 We show that P .t; u/ i is a rational function. We determine the rational functions which are obtained in that way for all the finite subgroups of SL 3 C.…”

Hilbert desingularizations for three dimensional canonical cyclic quotient singularities

“…By Theorem 6.1 in [7], it follows that arbitrary crepant toric resolution for a Gorenstein abelian quotient singularity is a Hilb-desingularization. Therefore, arbitrary crepant Fujiki-Oka resolution is Hilb-desingularization for any three dimensional semi-isolated Gorenstein quotient singularity.…”

“…• A crepant resolution is one of Hilb-desingularizations for toric quotient singularities in any dimension [7]. • For three dimensional terminal quotient singularities, Danilov [8] and Reid [19] introduce economic resolutions which is obtained by the sequence of weighted blow-ups.…”

New examples of compact manifolds with holonomy Spin(7)

“…If C n /G admits a crepant resolution, then the set of elements of age 1, σ 1 , is a minimal generating set for σ over Z. Theorem 6.2 gives us a necessary condition for a given cyclic quotient singularity to admit a crepant resolution. This condition is sufficient for all singularities of codimension 4 where the cyclic group has order less than 39 and is sufficient in all but 10 cases for quotient singularities of codimension 4 with cyclic group of order less than 100 [8]. (1, 1, 1, 1).…”