This is a survey of classical results of Grothendieck on vanishing cycles, such as the local monodromy theorem and his monodromy pairing for abelian varieties over local fields ([22, IX]). We discuss related current developments and questions. At the end, we include the proof of an unpublished result of Gabber giving a refined bound for the exponent of unipotence of the local monodromy for torsion coefficients.
Le présent texte est un exposé de résultats classiques de Grothendieck sur les cycles évanescents, tels que le théorème de monodromie locale et l’accouplement de monodromie pour les variétés abéliennes sur les corps locaux ([22, IX]). Nous présentons quelques développements récents et questions qui y sont liés. La dernière section est consacrée à la démonstration d’un résultat inédit de Gabber donnant une borne raffinée pour l’exposant d’unipotence de la monodromie locale pour des coefficients de torsion.
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Keywords: Étale cohomology, monodromy, Milnor fiber, nearby and vanishing cycles, alteration, hypercovering, semistable reduction, intersection complex, abelian scheme, Picard functor, Jacobian, Néron model, Picard–Lefschetz formula, $\ell $-adic sheaf
@article{AFST_2021_6_30_1_83_0, author = {Illusie, Luc}, title = {Grothendieck and vanishing cycles}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {83--115}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 30}, number = {1}, year = {2021}, doi = {10.5802/afst.1667}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/afst.1667/} }
TY - JOUR AU - Illusie, Luc TI - Grothendieck and vanishing cycles JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2021 SP - 83 EP - 115 VL - 30 IS - 1 PB - Université Paul Sabatier, Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1667/ DO - 10.5802/afst.1667 LA - en ID - AFST_2021_6_30_1_83_0 ER -
%0 Journal Article %A Illusie, Luc %T Grothendieck and vanishing cycles %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2021 %P 83-115 %V 30 %N 1 %I Université Paul Sabatier, Toulouse %U http://archive.numdam.org/articles/10.5802/afst.1667/ %R 10.5802/afst.1667 %G en %F AFST_2021_6_30_1_83_0
Illusie, Luc. Grothendieck and vanishing cycles. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 1, pp. 83-115. doi : 10.5802/afst.1667. http://archive.numdam.org/articles/10.5802/afst.1667/
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