Nearby cycles and characteristic classes of singular spaces

TL;DR

This paper explores the relationship between nearby cycles and characteristic classes of singular spaces, providing new insights into their behavior and differences in complex hypersurfaces using advanced algebraic and Hodge theoretic tools.

Contribution

It introduces a novel framework connecting nearby cycles with characteristic classes of singular spaces, enhancing understanding of their differences via vanishing cycles.

Findings

Describes the relation between nearby cycles and characteristic classes.

Provides a formula for differences between virtual and functorial classes.

Applies the theory to complex hypersurfaces with singularities.

Abstract

In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch- and motivic Chern-classes. As an application we get a description of the differences between the corresponding virtual and functorial characteristic classes of complex hypersurfaces in terms of vanishing cycles related to the singularities of the hypersurface.