Verdier specialization via weak factorization
type: Item
instance of: scholarly article
title: Verdier specialization via weak factorization
zbMATH Open document ID: 1278.14007
author: Paolo Aluffi
published in: Arkiv för Matematik
publication date: 2013-03-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.2483
review text: Let \(V\) be a variety and \(X\subset V\) a proper closed subscheme with the nonsingular complement. Consider a proper birational morphism \(f: W\rightarrow V\) such that \(W\) is nonsingular, \(D=f^{-1}(X)\) is a divisor with normal crossings and nonsingular components, and the restriction \(f|_{W\setminus D}\) is an isomorphism. The function \(\psi_{X,V}\) is defined to be the push-forward of \(\psi_{D,W},\) where \(\psi_{D,W}(p) = m\) if \(p\) is contained on a single component of \(D\) of multiplicity \(m,\) and \(\psi_{X,V}(p) = 0\) otherwise. The author discusses main properties of \(\psi_{X,V}\) in connection with some aspects of the theory of \textit{J.-L. Verdier} [Astérisque 82-83, 149--159 (1981; Zbl 0479.14013)]. For example, the function \(\psi_{X,V}\) is constructible and does not depend on the resolution in view of the weak factorization theorem. Next, \(\psi_{X,V}\) is compatible with the specialization of the Chern class of the complement \(V\setminus X\) in the Chow group, that is, \(c_{SM}(\psi_{X,V}) = i^*c_{SM}({\mathbf 1}_{V\setminus X}),\) where \(c_{SM}\) denotes the Chern-Schwartz-MacPherson class of constructible functions, \({\mathbf 1}_{V\setminus X} = {\mathbf 1}_V - {\mathbf 1}_X\) is the constant function, and \(i: X \hookrightarrow V\) is an effective Cartier divisor; this is a consequence of the intersection theory. Furthermore, the function \(\psi_{X,V}\) is birationally invariant in a certain weak sense, etc. The author also considers the corresponding motivic invariant \(\Psi_{X,V}\) defined in a suitable quotient of the Grothendieck ring of varieties. The class \(\Psi_{X,V}\) admits a natural monodromy decomposition; if \(X\) is the zero-set of a function on \(V\) then \(\Psi_{X,V}\) can be regarded as the motivic Milnor or nearby fiber in the sense [\textit{J. Denef} and \textit{F. Loeser}, in: Casacuberta, Carles (ed.) et al., 3rd European congress of mathematics (ECM), Barcelona, Spain, July 10-14, 2000. Volume I. Basel: Birkh…
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 32S45
zbMATH DE Number: 6149015
zbMATH Keywords: intersection theory
zbMATH Keywords: divisors with normal crossings
zbMATH Keywords: constructible functions
zbMATH Keywords: Chern classes
zbMATH Keywords: weak factorization
zbMATH Keywords: Cartier divisors
zbMATH Keywords: monodromy decomposition
zbMATH Keywords: motivic group
zbMATH Keywords: motivic Milnor fiber
Wikidata QID: Q60145521
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
OpenAlex ID: W3100193952
arXiv ID: 1009.2483
cites work: Torification and factorization of birational maps
cites work: La fonction zeta d'une monodromie
cites work: Chern classes for singular hypersurfaces
cites work: Q4411311
cites work: Q4658882
cites work: Modification systems and integration in their Chow groups
cites work: Limits of Chow groups, and a new construction of Chern-Schwartz-MacPherson classes
cites work: Q3447378
cites work: Chern classes of graph hypersurfaces and deletion-contraction
cites work: Feynman motives of banana graphs
cites work: Donaldson-Thomas type invariants via microlocal geometry
cites work: On motivic zeta functions and the motivic nearby fiber
cites work: Hirzebruch classes and motivic Chern classes.
cites work: HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES
cites work: Q3715247
cites work: Q4408042
cites work: Lefschetz numbers of iterates of the monodromy and truncated arcs
cites work: Q3328651
cites work: Macpherson's chern classes of singular algebraic varieties
cites work: Specialization of MacPherson's Chern classes.
cites work: Chern classes for singular algebraic varieties
cites work: A generalization of the Milnor number
cites work: Limits of tangent spaces to fibres and the \(w_ f\) condition
cites work: Q4526676
cites work: Q5514290
cites work: The monodromy conjecture for zeta functions associated to ideals in dimension two
cites work: Q3936838
cites work: Q3313978
DOI: 10.1007/S11512-011-0164-2
description: scientific article