Verdier specialization via weak factorization

Abstract

Let X⊂V be a closed embedding, with V∖X nonsingular. We define a constructible function ψX, V on X, agreeing with Verdier’s specialization of the constant function 1V 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 of the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich–Karu–Matsuki–Włodarczyk. The main property of ψX, V 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 ΨX, V in a motivic group. The function ψX, V and the corresponding Chern class cSM(ψX, V) and motivic aspect ΨX, V all have natural ‘monodromy’ decompositions, for any X⊂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.