The Topology of Surface Singularities

Abstract

We consider a reduced complex surface germ (X, p). We do not assume that X is normal at p, and so, the singular locus ( Σ, p) of (X, p) could be one dimensional. This text is devoted to the description of the topology of (X, p). By the conic structure theorem (see Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematical Studies 61 (1968), Princeton Univ. Press), (X, p) is homeomorphic to the cone on its link L X. First of all, for any good resolution ρ : (Y, E Y) → (X, 0) of (X, p), there exists a factorization through the normalization \(\nu : (\bar X,\bar p) \to (X,0 )\) (see H. Laufer, Normal two dimensional singularities, Ann. of Math. Studies 71, (1971), Princeton Univ. Press., Thm. 3.14). This is why we proceed in two steps. 1. When (X, p) a normal germ of surface, p is an isolated singular point and the link L X of (X, p) is a well defined differentiable three-manifold. Using the good minimal resolution of (X, p), L X is given as the boundary of a well defined plumbing (see Sect. 2.2) which has a negative definite intersection form (see Hirzebruch et al., Differentiable manifolds and quadratic forms, Math. Lecture Notes, vol 4 (1972), Dekker, New-York and Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), p. 299–344). 2. In Sect. 2.3, we use a suitably general morphism, \(\pi : (X,p) \to (\mathbb {C} ^2, 0)\), to describe the topology of a surface germ (X, p) which has a 1-dimensional singular locus ( Σ, p). We give a detailed description of the quotient morphism induced by the normalization ν on the link \(L_{\bar X}\) of \( (\bar X, \bar p)\) (see also Sect. 2.2 in Luengo-Pichon, Lê ‘s conjecture for cyclic covers, Séminaires et congrès 10, (2005), p. 163–190. Publications de la SMF, Ed. J.-P. Brasselet and T. Suwa). In Sect. 2.4, we give a detailed proof of the existence of a good resolution of a normal surface germ by the Hirzebruch-Jung method (Theorem 2.4.6). With this method a good resolution is obtained via an embedded resolution of the discriminant of π (see Friedrich Hirzebruch, Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen, Math. Ann. 126 (1953) p. 1–22). An example is given Sect. 2.6. An appendix (Sect. 2.5) is devoted to the topological study of lens spaces and to the description of the minimal resolution of quasi-ordinary singularities of surfaces. Section 2.5 provides the necessary background material to make the proof of Theorem 2.4.6 self-contained.