Verdier Specialization via Weak Factorization

Let X ⊂ V be a closed embedding, with V r 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 on the choice of resolution is obtained as a consequence of the weak factorization theorem of [AKMW02]. The main property of ψX,V is a compatibility with the specialization of the Chern class of the complement V r 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 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 .