Abstract
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–Włodarczyk. 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.
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Aluffi, P. Verdier specialization via weak factorization. Ark Mat 51, 1–28 (2013). https://doi.org/10.1007/s11512-011-0164-2
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DOI: https://doi.org/10.1007/s11512-011-0164-2