Grothendieck and vanishing cycles

Luc Illusie

doi:10.5802/afst.1667

Luc Illusie ¹

¹ Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 1, pp. 83-115.

Abstract
Abstract

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 At the end, we include the proof of an unpublished result of Gabber giving a refined bound for the exponent of unipotence of the local monodromy for torsion coefficients.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.

Received: 2018-10-12
Accepted: 2019-05-27
Published online: 2021-05-25

DOI: 10.5802/afst.1667

Classification: 01A65, 11F80, 11G10, 13D09, 14-03, 14D05, 14F20, 14G20, 14K30, 14H25, 14L05, 14L15, 32L55
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

Author's affiliations:

Luc Illusie ¹

¹ Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Luc Illusie. 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. https://afst.centre-mersenne.org/articles/10.5802/afst.1667/

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