Resolutions of Orbifold Singularities and Flows on the McKay Quiver

Abstract

Let $\Gamma$ be a finite group acting linearly on $\C^n$, 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 $X_\zeta$ (parameterised by $\zeta\in\Q^N$) which are partial resolutions of $\C^n/\Gamma$. In this article, it is shown the same $X_\zeta$ can be described as moduli spaces of representations of the McKay quiver associated to the action of $\Gamma$. It it shown that, for abelian groups, $X_\zeta$ 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 $X_\zeta $, its Euler number, and its singularities for any $\zeta$. The $\zeta$-parameter-space is thus partitioned into a finite disjoint union of cones inside which the biregular type of $X_\zeta$ remains constant. Finally, the example $\C^3/\Z_5$ (weights $1,2,3$) is worked out in detail, and figures of smooth and singular $X_\zeta$ and their corresponding flows are drawn. A further example of smooth crepant resolution $X_\zeta$ is drawn for the singularity ${1/11}(1,4,6)$.