Verdier specialization via weak factorization
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: Birkhäuser. Prog. Math. 201, 327--348 (2001; Zbl 1079.14003)], and so on. The paper contains a number of interesting examples explaining main features of the theory.
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