Perfect graphs and complex surface singularities with perfect local fundamental group

Abstract

In this paper we introduce the term to refer to those graphs which characterize resolutions of certain isolated singular points of complex surfaces. Using techniques for graphical evaluation of determinants, we reduce questions about perfect graphs to problems involving partial fraction representations of positive integers; the solutions to those Diophantine problems thus have interesting geometric interpretations. 1. Introduction and statement of results. In (5) Brieskorn gave the first examples of isolated singularities of complex n- varieties, n> 3, that are topologically non-singular (locally homeomorphic to the 2/2-ball) but analytically singular. Earlier Mumford (16) had shown that this is impossible in dimension 2. In this paper we pursue the natural analogue of the Brieskorn singularities for complex surfaces, namely those singular points xeX which are homologically non-singular in the sense of being locally homeomorphic to the cone on a homology 3-sphere. (The rational double point E8 is the most familiar example.) This condition is equivalent to the requirement that the local fundamental group of x in X be a perfect group (cf., for example, (16), (17), and (19), where the topic of classifying isolated two-dimensional singularities by the group-theoretic properties of the local fundamental group is introduced and developed). Let x be an isolated singularity of a normal complex surface X, and let p: X^>X be the minimal resolution of singularities . We will assume that the exceptional curve C=p~1(x)=\Jni = 1Ci is contractible, that each component Cf is non-singular rational, and that the components meet transversally with no triple intersections. In this case the topology of the singularity is completely determined by the weighted dual intersection graph Gp of the exceptional curve. In particular, the local fundamental group πx(x) can be computed directly from Gp in terms of generators and relations, by the technique of Mumford (16). Using this method it can be shown that πx(x) is perfect exactly when the intersection matrix { — Ci'C^ has determinant 1. Indeed, the following are necessary and sufficient conditions for a weighted graph G to be the dual graph of the minimal resolution of a normal complex surface singularity whose minimal resolution is normal (good) and whose local fundamental group is perfect: (a) G is a tree (a connected graph with no circuits). (b) Each weight wt is an integer >2. (c) The associated intersection matrix is positive definite with determinant 1. (Section 1 of (4) gives an elementary expository review of the geometry of complex surface