Enriched Relative Polar Curves and Discriminants

Let (f, g) be a pair of complex analytic functions on a singular analytic space X. We give “the correct” definition of the relative polar curve of (f, g), and we give a very formal generalization of Lê’s attaching result, which relates the relative polar curve to the relative cohomology of the Milnor fiber modulo a hyperplane slice. We also give the technical arguments which allow one to work with a derived category version of the discriminant and Cerf diagram of a pair of functions. From this, we derive a number of generalizations of results which are classically proved using the discriminant. In particular, we give applications to families of isolated “critical points”. 1 Lê’s Attaching Result and Our Previous Generalization Let U be an open neighborhood of the origin in C, and let f̃ : U → C be a complex analytic function. We assume that 0 ∈ V (f̃) := f̃(0). We let Σf̃ denote the critical locus of f̃ . In this paper, we describe an improvement/generalization of what is now a classic result in the study of singularities: the attaching result of Lê in [7], which tells one how many n-cells are attached, up to homotopy, to a hyperplane slice of the Milnor fiber of f̃ in order to obtain the Milnor fiber, Ff̃ ,0, of f̃ itself. However, first, we must discuss the relative polar curve. Fix a point p ∈ U . Let z0 denote a generic linear form on C, which, in fact, we take as the first coordinate function, after possibly performing a generic linear change of coordinates. In [4], [23], [7], [8], Hamm, Teissier, and Lê define and use the relative polar curve (of f̃ with respect to z0), Γ 1 f̃ ,z0 , to prove a number of topological results related to the Milnor fiber of hypersurface singularities. We shall recall some definitions and results here. We should mention that there are a number of different characterizations of the relative polar, all of which agree when z0 is sufficiently generic; below, we have selected what we consider the easiest way of describing the relative polar curve as a set, a scheme, and a cycle. The author would like to thank Tsukuba University, Tokyo Science University, Hokkaido University, and K. Takeuchi, M. Oka, and T. Ohmoto; Section 6 of this paper was written during a visit to these universities with the support of these mathematicians. AMS subject classifications 32B15, 32C35, 32C18, 32B10. keywords: polar curve, discriminant, Milnor fiber, nearby cycles, af condition.