Non-isolated hypersurface singularities and Lê cycles

Abstract

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 singularities. 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 Lê cycles and Lê numbers of non-isolated hypersurface singularities. Lecture 4 will explain the topological importance of Lê cycles and numbers, and then I will explain, informally, the relationship between the Lê cycles and the complex of sheaves of vanishing cycles.

Control by the lowest degree vanishing cycles

“…We continue with f as in the previous two sections; in particular, Σf ⊆ V (f ) and s := dim Σf = dim 0 Σf . The reader is referred to [4] and [5] for details of Lê cycles and Lê numbers, but we shall summarize needed properties here. Recall that the cycle Γ s f,z was defined in the previous section.…”