Non-isolated Hypersurface Singularities and L\^e Cycles

TL;DR

This paper discusses advanced topics in complex hypersurface singularities, focusing on L extsuperscript{e} cycles and numbers, and their topological significance, extending classical results to non-isolated singularities.

Contribution

It introduces new insights into L extsuperscript{e} cycles and numbers for non-isolated hypersurface singularities, linking them to sheaf complexes and topological invariants.

Findings

L extsuperscript{e} cycles are crucial for understanding non-isolated singularities.

L extsuperscript{e} numbers relate to the topology of hypersurface singularities.

The relationship between L extsuperscript{e} cycles and vanishing cycles is clarified.

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