Arkiv för Matematik

Verdier specialization via weak factorization

Paolo Aluffi

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Let XV be a closed embedding, with VX 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 VX. 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 XV 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.

Article information

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.

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