Verdier specialization via weak factorization

Paolo Aluffi Mathematics Department, Florida State University

K-Theory and Homology mathscidoc:1701.01019

Arkiv for Matematik, 51, (1), 1-28, 2010.10
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$V$.
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@inproceedings{paolo2010verdier,
  title={Verdier specialization via weak factorization},
  author={Paolo Aluffi},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203634866979038},
  booktitle={Arkiv for Matematik},
  volume={51},
  number={1},
  pages={1-28},
  year={2010},
}
Paolo Aluffi. Verdier specialization via weak factorization. 2010. Vol. 51. In Arkiv for Matematik. pp.1-28. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203634866979038.
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