Quadratic functions and smoothing surface singularities

Abstract

LET (X, 0) be an isolated complex analytic surface singularity which is reduced and irreducible. This paper is concerned with certain necessary conditions (X, 0) must satisfy in order that it admits a smoothingf: ( ?Z-, 0) + (A, 0); so,fis a flat morphism of analytic germs with smooth general fibre and X as special fibre. The general fibre of such an f may be represented by a Stein manifold of complex dimension 2, which is the interior of a compact 4manifold with boundary (M, dM); M is called the Milnorjbre of the smoothing. Moreover, there is a diffeomorphism of ZM onto a link L of (X, 0) (i.e. the intersection of X with a small sphere about 0 in an ambient space), which is well-defined up to isotopy. So, L may be regarded as the boundary of both M and a resolution .? of X. An important invariant of M is its middle homology group Hz(M) (which is free) equipped with the intersection product (which is symmetric). Note H2 (2) is also free and has a natural set of generators (the fundamental classes of the irreducible exceptional curves); the intersection pairing is negative-definite, but not necessarily even (even if X is a hypersurface). One would like to say as much as possible about the inner product lattice H2 (M) in terms of X or 2. For instance, if X is a Gorenstein singularity, then rk HI (M) = 0 [ll], the pairing on H2 (M) is even [38], and the Sylvester invariants (p,, , p + , p _ ) of the corresponding real pairing can be expressed in terms of x ( [41]: here, ~1 = p, + p+ + p- = rk H, (M)). Our principal contribution is the construction of natural and compatible “quadratic functions” on Hz (M), Hz(x), and H,(L), (where r means torsion subgroup). The construction and its properties are of a topological nature; we therefore need some topological discussion. If L is any compact 3-manifold, then there exists a natural symmetric non-degenerate linking pairing I: H, (L), x H, (L), + Q/Z defined geometrically [39]. Motivated by a result of Morgan-Sullivan [27], we prove that there is a finer object than 1. The singularity determines a homotopy class 9 of complex structures on 5‘ @ iw, (the tangent bundle of L plus a trivial bundle) whose first Chern class is torsion. Then (first part of Theorem 3.7) 9 determines a natural function 4 : H 1 (L), +Q/H such that 4(5*+9)-4(5)-4(rl)=1(5,~). Since 1 is bilinear, we call 4 a quadratic function; if furthermore 4 (no = n2 4 (<) (n E Z), then 4 is called a quadratic form. The function 4 derives its main interest from a universal property with respect to the intersection pairings of compact almost-complex 4-manifolds which have