On the existence of crepant resolutions of Gorenstein Abelian quotient singularities in dimensions \geq 4

Abstract

For which finite subgroups G of SL(r,C), r \geq 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.

Version v2 (April 2006) contains minor changes and corrections of some typos. A shorter version of the paper appears in the Proceedings of the Conference "Algebraic and Geometric Combinatorics" (held in Anogia, Crete, 20-26 August 2005)

References in corpus (11)

• Lectures on 0/1-polytopes
• McKay equivalence for symplectic resolutions of singularities
• A Monograph on the Classification of the Discrete Subgroups of SU(4)
• How to calculate A-Hilb C^3
• Gorenstein Quotient Singularities of Monomial Type in Dimension Three
• A correspondence dual to McKay's
• Birational geometry in codimension 2 of symplectic resolutions
• On the Computation of Hilbert Bases and Extreme Rays of Cones
• On a series of Gorenstein cyclic quotient singularities admitting a unique projective crepant resolution
• Symplectic Resolutions for Quotient Singularities
• Crepant Resolutions of Gorenstein Toric Singularities and Upper Bound Theorem

Cited by in corpus (7)

• ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds
• On genus one mirror symmetry in higher dimensions and the BCOV conjectures
• New Examples of Compact Manifolds with Holonomy Spin(7)
• Crepant Property of Fujiki-Oka Resolutions for Gorenstein Abelian Quotient Singularities
• Crepant resolutions and Hilb^G(C^4) for certain abelian subgroups for SL(4,C)
• Hilbert desingularizations for three dimensional canonical cyclic quotient singularities
• Quotients of the Dwork Pencil