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We give a new proof—not using resolution of singularities—of a formula of Denef and the second author expressing 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. Our proof uses -adic cohomology of non-archimedean spaces, motivic integration and the Lefschetz fixed point formula for finite order automorphisms. We also consider a generalization due to Nicaise and Sebag and at the end of the paper we discuss connections with the motivic Serre invariant and the motivic Milnor fiber.
Nous donnons une nouvelle preuve — n'utilisant pas la résolution des singularités — d'une formule de Denef et du second auteur exprimant le nombre de Lefschetz des itérés de la monodromie d'une fonction sur une variété algébrique complexe en fonction de la caractéristique d'Euler d'un espace d'arcs tronqués. Notre preuve utilise la cohomologie -adique des espaces non-archimédiens, l'intégration motivique, ainsi que la formule des points fixes de Lefschetz pour les automorphismes d'ordre fini. Nous considérons également une généralisation due à Nicaise et Sebag et la fin de l'article est consacrée aux relations avec l'invariant de Serre motivique et la fibre de Milnor motivique.
@article{ASENS_2015__48_2_313_0,
author = {Hrushovski, Ehud and Loeser, Fran\c{c}ois},
title = {Monodromy and the {Lefschetz} fixed point formula},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {313--349},
publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
volume = {Ser. 4, 48},
number = {2},
year = {2015},
doi = {10.24033/asens.2246},
mrnumber = {3346173},
zbl = {1400.14015},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.24033/asens.2246/}
}
TY - JOUR AU - Hrushovski, Ehud AU - Loeser, François TI - Monodromy and the Lefschetz fixed point formula JO - Annales scientifiques de l'École Normale Supérieure PY - 2015 SP - 313 EP - 349 VL - 48 IS - 2 PB - Société Mathématique de France. Tous droits réservés UR - http://geodesic.mathdoc.fr/articles/10.24033/asens.2246/ DO - 10.24033/asens.2246 LA - en ID - ASENS_2015__48_2_313_0 ER -
%0 Journal Article %A Hrushovski, Ehud %A Loeser, François %T Monodromy and the Lefschetz fixed point formula %J Annales scientifiques de l'École Normale Supérieure %D 2015 %P 313-349 %V 48 %N 2 %I Société Mathématique de France. Tous droits réservés %U http://geodesic.mathdoc.fr/articles/10.24033/asens.2246/ %R 10.24033/asens.2246 %G en %F ASENS_2015__48_2_313_0
Hrushovski, Ehud; Loeser, François. Monodromy and the Lefschetz fixed point formula. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 2, pp. 313-349. doi : 10.24033/asens.2246. http://geodesic.mathdoc.fr/articles/10.24033/asens.2246/
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