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Lefschetz numbers of iterates of the monodromy and truncated arcs

De
10 pages
Topology 41 (2002) 1031–1040 Lefschetz numbers of iterates of the monodromy and truncated arcs Jan Denef a ; 1 , Fran'cois Loeser b; ?; 2 a Department of Mathematics, University of Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium b Departement de Mathematiques et Applications, Ecole Normale Superieure, 45 Rue d'Ulm, 75230 Paris Cedex 05, France Received 14 January 2000; accepted 26 February 2001 Abstract We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. We also construct a canonical representative of the Milnor /bre in a suitable monodromic Grothendieck group. ? 2002 Elsevier Science Ltd. All rights reserved. MSC: 14B05; 14J17; 32S25; 32S55 Keywords: Monodromy; Milnor /bre; Arcs; Singularities; Lefschetz numbers; Motivic integration; Monodromic Grothendieck group 1. Introduction Let X be a smooth complex algebraic variety and let f : X ? C be a non constant morphism of complex algebraic varieties. We /x a smooth metric on X . Let x be a point of f ?1 (0).

  • characteristic zero

  • group generated

  • involve truncated

  • arcs modulo

  • arc

  • taking euler characteristic

  • truncated arcs

  • grothendieck group


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Lefschetz
Topology 41 (2002) 1031–1040
numbers
of iterates of the truncated arcs
www.elsevier.com/locate/top
monodromy
and
a;1 b;;2 Jan Denef , Franc'ois Loeser a Department of Mathematics, University of Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium b DepartementdeMathematiquesetApplications,EcoleNormaleSuperieure,45RuedUlm, 75230 Paris Cedex 05, France Received 14 January 2000; accepted 26 February 2001
Abstract
We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. We also construct a canonical representative of the Milnor bre in a suitable monodromic Grothendieck group.?2002 Elsevier Science Ltd. All rights reserved.
MSC:14B05; 14J17; 32S25; 32S55 Keywords:Monodromy; Milnor bre; Arcs; Singularities; Lefschetz numbers; Motivic integration; Monodromic Grothendieck group
1. Introduction
LetXbe a smooth complex algebraic variety and letf:XCbe a non constant morphism 1 of complex algebraic varieties. We x a smooth metric onX. Letxbe a point off(0). We set × −1× × X:=B(x; )f(D), withB(x; ) the open ball of radiuscentered atx=D ;  andD \ {0}, × withDthe open disk of radiuscentered at 0. For 0¡ 1, the restriction offtoX ;  × is a locally trivial bration – called the Milnor bration – ontoDwith berFx, the Milnor
Corresponding author. E-mail addresses:jan.denef@wis.kuleuven.ac.be (J. Denef), francois.loeser@ens.fr (F. Loeser). 1 URL: http==www.wis.kuleuven.ac.be=wis=algebra=denef.html. 2 URL: http==www.dma.ens.fr=loeser=.
0040-9383/02/$ - see front matter?2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 0 - 9 3 8 3 ( 0 1 ) 0 0 0 1 6 - 7
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