A characteristic number for links of surface singularities

Abstract

Milnor and Thurston [MT] define a characteristic number of a closed orientable 3-manifold M to be a real-valued topological invariant (r(M) such that: if (p(M) is defined, and M is a k-sheeted covering of M, then (p(M) is defined and equal to k(p(M) . This multiplicative property is clear for the Euler characteristic, but of course this is 0 for a 3-manifold. If M has an appropriate Riemannian metric (e.g., hyperbolic, or with all sectional curvatures + 1), then the volume is a characteristic number. Milnor and Thurston introduce several characteristic numbers; one is defined for all 3-manifolds, is positive on a fixed hyperbolic manifold, and has the additional property that if M -N is a general map of degree k, then ( (M) > I kk I (N). Another characteristic number is the simplicial volume introduced by Gromov [G]. In this paper, we introduce a characteristic number for each link of a complex surface singularity. It is a nonnegative rational number, is 0 only in wellunderstood cases, is computable from any resolution dual graph (=plumbing diagram), and has in addition the submultiplicative property above for degree k maps arising from morphisms of the singularities themselves. This is the only characteristic number for links which we know of that can be computed from the graph. We call this invariant -P. P, since it is the negative of the self-intersection number of a (rational) cycle on a complex surface (a resolution of the singularity). The definition arose from our work on the generalized Miyaoka inequality for normal surfaces [W3] (this term appeared in the inequality). Another ingredient is the notion of Zariski decomposition of a line bundle or divisor on a surface, especially on a resolution of a surface singularity, as in Sakai [S]. Let (X, o) be the germ of a normal complex surface singularity (necessarily isolated) with X contractible, and AX M the link of X. M is a closed, connected, orientable 3-manifold. Let (X, E) -(X, o) be a good resolution; hence, the inverse image E = U Ei of o is the union of nonsingular curves, intersecting transversally, no three through a point. The resolution dual graph