Singularities at infinity and motivic integration
type: Item
instance of: scholarly article
zbMATH Open document ID: 1266.14012
author: Michel Raibaut
publication date: 2012-05-10T00:00:00Z
full work available at URL: smf bull 140 51-100.php
Mathematics Subject Classification ID: 14E18
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 32S35
Mathematics Subject Classification ID: 32S55
zbMATH DE Number: 6032259
zbMATH Keywords: Grothendieck ring
zbMATH Keywords: motivic Milnor fiber
zbMATH Keywords: Newton polyhedron
zbMATH Keywords: motivic bifurcation set
MaRDI profile type: Publication
OpenAlex ID: W2097792816
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DOI: 10.24033/BSMF.2624
title: Singularities at infinity and motivic integration
published in: Bulletin de la Société Mathématique de France
review text: If given a smooth complex variety \(X\) and a non constant morphism \(f : X \to \mathbb A^1\) there exists a finite set \(S \subset \mathbb A^1\), called the bifurcation set, such that the restriction of \(f\) on \(X \setminus f ^{-1} (S)\) is a locally trivial topological fibration; the critical values of \(f\) belong to \(S\). The cohomology groups with compact support \(H_c^*(f^{-1} (t),\mathbb Q)\) have a natural mixed Hodge structure ([\textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 44, 5--77 (1974; Zbl 0237.14003)]). Then could be defined the limit mixed Hodge structure ([\textit{J. Steenbrink} and \textit{S. Zucker}, Invent. Math. 80, 489--542 (1985; Zbl 0626.14007)]) whose spectrum is an invariant of \(f\), called spectrum at infinity ([\textit{A. Dimca}, NATO Sci. Ser. II, Math. Phys. Chem. 21, 257--278 (2001; Zbl 0993.32017)], [\textit{R. García López} and \textit{A. Némethi}, Compos. Math. 100, No. 2, 205--231 (1996; Zbl 0855.32016)], [\textit{R. García López} and \textit{A. Némethi}, Compos. Math. 115, No. 1, 1--20 (1999; Zbl 0947.32014)], [\textit{A. Némethi} and \textit{C. Sabbah}, Abh. Math. Semin. Univ. Hamb. 69, 25--35 (1999; Zbl 0973.32014)]). In the article under review are introduced and studied the motivic counterparts of these and other related with them notions.NEWLINENEWLINEFor any morphism \(f : X \to\mathbb A^1\) over a field \(k\) of characteristic 0, and any closed point \(x\inf^{-1}(0)\), \textit{J. Denef} and \textit{F. Loeser} [J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010); Prog. Math. 201, 327--348 (2001; Zbl 1079.14003)] introduced the motivic Milnor fiber \(S_{f,x}\) in the Grothendieck ring of varieties with a \(\mathbb G_m\) action. It specializes in the Grothendieck ring of Hodge structures with a quasi unipotent endomorphism \(K_0 (SH^{mon})\) to the class of the limit mixed Hodge structure of the Milnor fiber of \(f\) at \(x\). \textit{F. Bittner} [Math. Z. 249, No. 1, 63--83 (2005; Zbl 1085.14020)] and \…
reviewed by: Peter Petrov
description: scientific article; zbMATH DE number 6032259