Verdier specialization via weak factorization

TL;DR

This paper introduces a new definition of Verdier's specialization for constructible functions on a closed embedding, ensuring independence from resolution choices and compatibility with Chern class specialization, with applications to motivic groups and monodromy decompositions.

Contribution

It provides a resolution-independent, constructible function-based definition of Verdier's specialization that aligns with intersection theory and extends to motivic and monodromy contexts.

Findings

Defines a constructible specialization function compatible with Chern class specialization.

Ensures independence from resolution choices via the weak factorization theorem.

Recovers Verdier's original result for zero-locus functions and relates to Behrend's constructible function.

Abstract

Let X in V be a closed embedding, with V - X nonsingular. We define a constructible function on X, agreeing with Verdier's specialization of the constant function 1 when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence on the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich et al. The main property of the specialization function is a compatibility with the specialization of the Chern class of the complement V-X. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier's result when X is the zero-locus of a function on V. Our definition has a straightforward counterpart in a motivic group. The specialization function and the corresponding Chern class and motivic aspect all have natural `monodromy' decompositions, for for any X in V as above. The definition also yields an expression for Kai Behrend's constructible function when applied to (the singularity subscheme of) the zero-locus of a function on V.