Algebraic geometry
Computing zeta functions on log smooth models
[Calcul de fonctions zêta à partir de modèles log lisses]

Présenté par : Claire Voisin

Bultot, Emmanuel1
1 KU Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Heverlee, Belgium
Comptes Rendus. Mathématique, Tome 353 (2015) no. 3, pp. 261-264.

Nous établissons une formule pour la série volume de Poincaré d'un schéma log lisse. Ceci nous fournit en particulier une nouvelle expression et un ensemble réduit de candidats pôles pour la fonction zêta motivique d'une singularité d'hypersurface et d'une dégénération de variétés de Calabi–Yau.

We establish a formula for the volume Poincaré series of a log smooth scheme. This yields in particular a new expression and a smaller set of candidate poles for the motivic zeta function of a hypersurface singularity and of a degeneration of Calabi–Yau varieties.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.11.014
Bultot, Emmanuel 1

1 KU Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Heverlee, Belgium 
@article{CRMATH_2015__353_3_261_0,
     author = {Bultot, Emmanuel},
     title = {Computing zeta functions on log smooth models},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {261--264},
     publisher = {Elsevier},
     volume = {353},
     number = {3},
     year = {2015},
     doi = {10.1016/j.crma.2014.11.014},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2014.11.014/}
}
TY  - JOUR
AU  - Bultot, Emmanuel
TI  - Computing zeta functions on log smooth models
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 261
EP  - 264
VL  - 353
IS  - 3
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2014.11.014/
DO  - 10.1016/j.crma.2014.11.014
LA  - en
ID  - CRMATH_2015__353_3_261_0
ER  - 
%0 Journal Article
%A Bultot, Emmanuel
%T Computing zeta functions on log smooth models
%J Comptes Rendus. Mathématique
%D 2015
%P 261-264
%V 353
%N 3
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2014.11.014/
%R 10.1016/j.crma.2014.11.014
%G en
%F CRMATH_2015__353_3_261_0
Bultot, Emmanuel. Computing zeta functions on log smooth models. Comptes Rendus. Mathématique, Tome 353 (2015) no. 3, pp. 261-264. doi : 10.1016/j.crma.2014.11.014. http://archive.numdam.org/articles/10.1016/j.crma.2014.11.014/

[1] Denef, Jan; Loeser, François Geometry on arc spaces of algebraic varieties, Barcelona, 2000 (Progr. Math.), Volume vol. 201, Birkhäuser, Basel, Switzerland (2001), pp. 327-348

[2] Gross, Mark; Siebert, Bernd An invitation to toric degenerations, Surveys in Differential Geometry, Volume XVI: Geometry of Special Holonomy and Related Topics, Surv. Differ. Geom., vol. 16, Int. Press, Somerville, MA, USA, 2011, pp. 43-78

[3] Guibert, Gil Espaces d'arcs et invariants d'Alexander, Comment. Math. Helv., Volume 77 (2002) no. 4, pp. 783-820

[4] Halle, Lars Halvard; Nicaise, Johannes Motivic zeta functions of Abelian varieties, and the monodromy conjecture, Adv. Math., Volume 227 (2011) no. 1, pp. 610-653

[5] Halle, Lars Halvard; Nicaise, Johannes Motivic zeta functions for degenerations of Abelian varieties and Calabi–Yau varieties, Zeta Functions in Algebra and Geometry, Contemp. Math., vol. 566, Amer. Math. Soc., Providence, RI, USA, 2012, pp. 233-259

[6] Kato, Kazuya Logarithmic structures of Fontaine–Illusie, Baltimore, MD, USA, 1988, Johns Hopkins Univ. Press, Baltimore, MD, USA (1989), pp. 191-224

[7] Kato, Kazuya Toric singularities, Amer. J. Math., Volume 116 (1994) no. 5, pp. 1073-1099

[8] Nicaise, Johannes; Sebag, Julien Motivic Serre invariants, ramification, and the analytic Milnor fiber, Invent. Math., Volume 168 (2007) no. 1, pp. 133-173

[9] Nicaise, Johannes; Sebag, Julien Motivic invariants of rigid varieties, and applications to complex singularities, Motivic Integration and Its Interactions with Model Theory and Non-Archimedean Geometry, Volume I, Lond. Math. Soc. Lect. Note Ser., vol. 383, Cambridge Univ. Press, Cambridge, UK, 2011, pp. 244-304

Cité par Sources :