1996/10/04 by Alexander V. Sardo-Infirri, Sardo-Infirri, Alexander V
Mathematics · #14M25 (Primary) 05C35 #32S45 (Secondary) #90C08 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.alg-geom/9610005
openalex publication_date 1996/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be a finite group acting linearly on \Cn, freely outside the origin. In previous work a generalisation of Kronheimer's construction of moduli of Hermitian-Yang-Mills bundles with certain invariance properties was given. This produced varieties Xζ (parameterised by ζ∈\QN) which are partial resolutions of \Cn/Γ. In this article, it is shown the same Xζ can be described as moduli spaces of representations of the McKay quiver associated to the action of Γ. It it shown that, for abelian groups, Xζ are toric varieties defined by convex polyhedra which are the solution sets for a generalisation of the transportation problem on the McKay quiver. The generalised transportation problem is solved for an arbitrary quiver to give a description of the extreme points, faces, and tangent cones to the solution polyhedra in terms of certain distinguished trees in the quiver. Applied the McKay quiver, this gives an explicit procedure for calculating Xζ, its Euler number, and its singularities for any ζ. The ζ-parameter-space is thus partitioned into a finite disjoint union of cones inside which the biregular type of Xζ remains constant. Finally, the example \C3/\Z5 (weights 1,2,3) is worked out in detail, and figures of smooth and singular Xζ and their corresponding flows are drawn. A further example of smooth crepant resolution Xζ is drawn for the singularity 1/11(1,4,6).