NON-ISOLATED HYPERSURFACE SINGULARITIES AND LE ̂ CYCLES

Abstract. In this series of lectures, I will discuss results for complex hypersurfaces with non-isolated singularities. In Lecture 1, I will review basic definitions and results on complex hypersurfaces, and then present classical material on the Milnor fiber and fibration. In Lecture 2, I will present basic results from Morse theory, and use them to prove some results about complex hypersurfaces, including a proof of Lê’s attaching result for Milnor fibers of non-isolated hypersurface sin-gularities. This will include defining the relative polar curve. Lecture 3 will begin with a discussion of intersection cycles for proper intersections inside a complex manifold, and then move on to definitions and basic results on Le ̂ cycles and Le ̂ numbers of non-isolated hypersur-face singularities. Lecture 4 will explain the topological importance of Le ̂ cycles and numbers, and then I will explain, informally, the relationship between the Le ̂ cycles and the complex of sheaves of vanishing cycles. 1. Lecture 1: Topology of Hypersurfaces and the Milnor fibration Suppose that U is an open subset of Cn+1; we use (z0,..., zn) for coordinates. Consider a complex analytic (i.e., holomorphic) function f: U → C which is not locally constant. Then, the hypersurface V (f) defined by f is the purely n-dimensional complex analytic space defined by the vanishing of f, i.e., V (f): = {x ∈ U | f(x) = 0}. To be assured that V (f) is not empty and to have a convenient point in V (f), one frequently assumes that 0 ∈ V (f), i.e., that f(0) = 0. This assumption is frequently included in specifying the function, e.g., we frequently write f: (U,0) → (C, 0). Near each point x ∈ V (f), we are interested in the local topology of how V (f) is embedded in U. This is question of how to describe the local, ambient topological-type of V (f) at each point. A critical point of f is a point x ∈ U at which all of the complex partial derivatives of f vanish. The critical locus of f is the set of critical points of f, and is denoted by Σf, i.e.