Arkiv för Matematik
- Ark. Mat.
- Volume 51, Number 1 (2013), 1-28.
Verdier specialization via weak factorization
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.
Ark. Mat., Volume 51, Number 1 (2013), 1-28.
Received: 15 October 2010
First available in Project Euclid: 31 January 2017
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Aluffi, Paolo. Verdier specialization via weak factorization. Ark. Mat. 51 (2013), no. 1, 1--28. doi:10.1007/s11512-011-0164-2. https://projecteuclid.org/euclid.afm/1485907202