The signature of smoothings of complex surface singularities

Abstract

Let f : (1~3, 0)-"~(t~, 0) be the germ of a complex analytic function with an isolated critical point at the origin. For e > 0 suitably small and 6 yet smaller, the space V ' = f l ( 6 ) ~ D , (where D~ denotes the closed disk of radius e about 0) is a real oriented four-manifold with boundary whose diffeomorphism type depends only on f It has been proved that V' has the homotopy type of a wedge of two-spheres; the number p o f two-spheres is readily computable. Recently an interesting formula for g was given in terms of analytic invariants of a resolution of the singularity at 0 of the complex surface f-1(0) [13]. This formula is proved by applying the Riemann-Roch theorem to the projective completions o f f l ( 0 ) and fi (6) , then canceling terms coming from the parts away from the origin. The purpose of this paper is to find a similar formula for the signature of the intersection pairing on the two-dimensional homology of the manifold V', using the Hirzebruch signature theorem instead of the Riemann-Roch theorem. Various other signature formulas are known, in higher dimensions as well as in dimension two. For f (x , y, z) of the form g(x, y) + z 2, the intersection pairing of V' is the same as a symmetrized Seifert matrix of the compound torus link {g(x, y )=0}~S 3. There is a simple formula for the signature of the symmetrized Seifert matrix of a compound link of one component [20]; hence if g-1(0) is irreducible, there is a simple formula for the signature of V' in terms of the Puiseux pairs ofg. If g1(0) has several branches at the origin, it is possible to find a Seifert matrix for the link defined by g and compute the signature [17], but this process is tedious for all but the simplest links. Formulas also exist for the signature when f(x, y, z) is of the type x a + yb + z c [10], or when f is weighted homogeneous [22]. There is in addition a formula for the signature in terms of mixed Hodge structure [233. The genus (or geometric genus) of the singularity f 1(0) (assumed Stein) is the dimension of HI(V,, C~), where ~" is a resolution of f-1(0). By combining the formulas for the signature and the number # it is possible to show that twice the genus of the singularity f 1(0) is equal to the number of positive plus the number of