Verdier specialization via weak factorization

Let X?V be a closed embedding, with V?X nonsingular. We define a constructible function ψ _X,V on X, agreeing with Verdier’s specialization of the constant function 1 _V 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–W?odarczyk. The main property of ψ _X,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 Ψ _X,V in a motivic group. The function ψ _X,V and the corresponding Chern class c _SM(ψ _X,V ) and motivic aspect Ψ _X,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.