Grothendieck and vanishing cycles
type: Item
instance of: scholarly article
title: Grothendieck and vanishing cycles
zbMATH Open document ID: 1473.14001
author: Luc Illusie
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2021-08-17T00:00:00Z
review text: The paper is a survey of classical results of Grothendieck on vanishing cycles. Consider a regular, proper and flat curve \(X\) over a strictly local trait \(S=(S,s,\eta )\), whose generic fiber is smooth and whose reduced special fiber is a divisor with normal crossings. One can analyse the difference between the (étale) cohomology of the special fiber \(H^*(X_s)\) and that of the generic geometric fiber \(H^*(X_{\eta})\), the coefficients ring being \(\mathbb{Z}_{\ell}\), \(\ell\) a prime number invertible on \(S\). Assuming that the action of the inertia group \(I\) on \(H^*(X_{\bar{\eta}})\) is tame, one shows that the defect of the specialization map \(H^*(X_s)\rightarrow H^*(X_{\bar{\eta}})\) is an isomorphism controlled by the vanishing cycles groups. This construction leads to Grothendieck's local monodromy theorem and his monodromy pairing for abelian varieties over local fields. The author discusses related current developments and questions and includes the proof of an unpublished result of Gabber giving a refined bound for the exponent of unipotence of the local monodromy for torsion coefficients.
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14-03
Mathematics Subject Classification ID: 01A65
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 13D09
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14K30
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14L15
zbMATH DE Number: 7382715
zbMATH Keywords: étale cohomology
zbMATH Keywords: monodromy
zbMATH Keywords: Milnor fiber
zbMATH Keywords: nearby and vanishing cycles
zbMATH Keywords: alteration
zbMATH Keywords: hypercovering
zbMATH Keywords: semistable reduction
zbMATH Keywords: intersection complex
zbMATH Keywords: abelian scheme
zbMATH Keywords: Picard functor
zbMATH Keywords: Jacobian
zbMATH Keywords: Néron model
zbMATH Keywords: Picard-Lefschetz formula
zbMATH Keywords: \(\ell\)-adic sheaf
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: afst.1667
OpenAlex ID: W3165792560
cites work: Q4112660
cites work: Seminar of algebraic geometry du Bois-Marie 1963--1964. Topos theory and étale cohomology of schemes (SGA 4). Vol. 1: Topos theory. Exp. I--IV
cites work: Degenerate fibres and stable reduction of curves
cites work: Dualité sur un corps local à corps résiduel algébriquement clos
cites work: Constructible sheaves are holonomic
cites work: Q4274265
cites work: Q4640173
cites work: On perfectness of Grothendieck's pairing for the \(l\)-parts of component groups
cites work: Q4944268
cites work: Q4349379
cites work: Rational points over finite fields for regular models of algebraic varieties of Hodge type \(\geq 1\)
cites work: Component groups of abelian varieties and Grothendieck's duality conjecture
cites work: Grothendieck's pairing on component groups of Jacobians.
cites work: Néron Models
cites work: Correspondence Grothendieck-Serre
cites work: Q5623820
cites work: Théorie de Hodge. II. (Hodge theory. II)
cites work: La conjecture de Weil. I
cites work: Théorie de Hodge. III
cites work: Q4119314
cites work: La conjecture de Weil. II
cites work: The irreducibility of the space of curves of a given genus
cites work: Etale Cohomology Theory
cites work: Q4407869
cites work: Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale
cites work: Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems
cites work: Q5634092
cites work: Q4126449
cites work: Abstract vanishing cycle theory
cites work: Q4026192
cites work: Q4325441
cites work: Q4539743
cites work: Q4407871
cites work: Q3154992
cites work: Miscellany on traces in \(\ell\)-adic cohomology: a survey
cites work: Q2945028
cites work: From Pierre Deligne's secret garden: looking back at some of his letters
cites work: Q5693279
cites work: Weight-monodromy conjecture for \(p\)-adically uniformized varieties
cites work: Weight-monodromy conjecture over equal characteristic local fields

Grothendieck and vanishing cycles (cites work)
cites work: Crystalline Dieudonné module theory via formal and rigid geometry

Grothendieck and vanishing cycles (cites work)
cites work: Smoothness, semi-stability and alterations

Grothendieck and vanishing cycles (cites work)
cites work: Q3682590

Grothendieck and vanishing cycles (cites work)
cites work: Duality and nearby cycles over general bases

Grothendieck and vanishing cycles (cites work)
cites work: K-theory, arithmetic and geometry. Seminar, Moscow University, 1984-1986

Grothendieck and vanishing cycles (cites work)
cites work: Modular curves and the Eisenstein ideal

Grothendieck and vanishing cycles (cites work)
cites work: Duality theorems for Néron models

Grothendieck and vanishing cycles (cites work)
cites work: Singular Points of Complex Hypersurfaces. (AM-61)

Grothendieck and vanishing cycles (cites work)
cites work: Q5619906

Grothendieck and vanishing cycles (cites work)
cites work: Q4392799

Grothendieck and vanishing cycles (cites work)
cites work: Degeneration of l -adic weight spectral sequences

Grothendieck and vanishing cycles (cites work)
cites work: Modifications et cycles proches sur une base generale

Grothendieck and vanishing cycles (cites work)
cites work: Über die lokale Zetafunktion von Shimuravarietaeten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik

Grothendieck and vanishing cycles (cites work)
cites work: Faisceaux amples sur les schémas en groupes et les espaces homogènes

Grothendieck and vanishing cycles (cites work)
cites work: Spécialisation du foncteur de Picard

Grothendieck and vanishing cycles (cites work)
cites work: Q5628265

Grothendieck and vanishing cycles (cites work)
cites work: On modular representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms

Grothendieck and vanishing cycles (cites work)
cites work: Vanishing Cycles and Geometry of Curves Over a Discrete Valuation Ring

Grothendieck and vanishing cycles (cites work)
cites work: WEIGHT SPECTRAL SEQUENCES AND INDEPENDENCE OF $\ell$

Grothendieck and vanishing cycles (cites work)
cites work: The characteristic cycle and the singular support of a constructible sheaf

Grothendieck and vanishing cycles (cites work)
cites work: Wild ramification and the cotangent bundle

Grothendieck and vanishing cycles (cites work)
cites work: Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for \(\mathrm{SL}_2\)

Grothendieck and vanishing cycles (cites work)
cites work: Perfectoid spaces

Grothendieck and vanishing cycles (cites work)
cites work: Q3277020

Grothendieck and vanishing cycles (cites work)
cites work: Good reduction of abelian varieties

Grothendieck and vanishing cycles (cites work)
cites work: Limits of Hodge structures

Grothendieck and vanishing cycles (cites work)
cites work: Parity and symmetry in intersection and ordinary cohomology

Grothendieck and vanishing cycles (cites work)
cites work: Néron models of 1-motives and duality

Grothendieck and vanishing cycles (cites work)
cites work: Grothendieck's pairing on Néron component groups: Galois descent from the semistable case

Grothendieck and vanishing cycles (cites work)
cites work: Q5539560

Grothendieck and vanishing cycles (cites work)
cites work: Stable modification of relative curves

Grothendieck and vanishing cycles (cites work)
cites work: A uniform bound for the order of monodromy

Grothendieck and vanishing cycles (cites work)
cites work: Q4339900

Grothendieck and vanishing cycles (DOI)
DOI: 10.5802/AFST.1667