Nonlinear Sections of Nonisolated Complete Intersections

By a series of discoveries during the past thirty years, a distinguished list of researchers have provided us with a marvelous vista of singularity theory as it applies to isolated singularities, especially isolated complete intersection singularities. This began with Milnor’s seminal monograph on isolated hypersurfaces singularities [Mi], which introduced as a principal tool in the study of isolated singularities the Milnor fibration of the singularity. The basic results were extended by Hamm [Ha] to isolated complete intersection singularities (ICIS). There has followed a succession of revelations concerning the topology, local geometry, and deformation theory of ICIS using: De Rham cohomology and GaussManin connection, intersection pairing and monodromy, mixed Hodge structures and spectrum, structure of discriminants, equsingularity via multiplicities; and deformation theory. We refer to e.g. [Lo] and [?]vol 2]AVG where many of these results are presented. If we were to seek a comparable view of the more complicated nonisolated singularities, then the results for ICIS provide a virtual “wish list”of types of results to be obtained. However, now the vista is considerably clouded, lacking many of the details so apparent for ICIS, although revealing general features via techniques involving stratification theory, resolution of singularities, etc. A sample of the kinds of questions involving nonisolated singularities which we will consider involve e.g.: topology of complements of hyperplane arrangements, the topology of boundary singularities of complete intersections, critical points of functions f1 1 · · · fλr r appearing in hypergeometric functions, minimum Ae–codimenson for germs f : C| n, 0 → C| p, 0 in a given contact class, de Rham cohomology of highly singular spaces, higher multiplicities à la Teissier for nonisolated singularities together with Buchbaum–Rim multiplicities of modules; as well as properties of discriminants and bifurcation sets for various notions of equivalence. These examples share no obvious common feature except ultimately they concern properties of highly nonisolated singular spaces. ∗ Partially supported by a grant from the National Science Foundation