Troisième groupe de cohomologie non ramifiée d'un solide cubique sur un corps de fonctions d'une variable

Jean Louis Colliot-Thélène; Alena Pirutka

Troisième groupe de cohomologie non ramifiée d'un solide cubique sur un corps de fonctions d'une variable

Translated title of the contribution: Third unramified cohomology group of a cubic threefold over a function field in one variable

Jean Louis Colliot-Thélène, Alena Pirutka

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that the third unramified cohomology group of a smooth cubic threefold over the function field of a complex curve vanishes. For this, we combine a method of C. Voisin with Galois descent on the codimension 2 Chow group. As a corollary, we show that the integral Hodge conjecture holds for degree 4 classes on smooth projective fourfolds equipped with a fibration over a curve, the generic fibre of which is a smooth cubic threefold, with arbitrary singularities on the special fibres.

Translated title of the contribution	Third unramified cohomology group of a cubic threefold over a function field in one variable
Original language	French
Article number	24
Journal	Epijournal de Geometrie Algebrique
Volume	2
State	Published - 2018

Keywords

Chow groups
Codimension 2 cycles
Family of cubic hypersurfaces
Integral Hodge conjecture
Intermediate jacobian
Unramified cohomology

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

Cite this

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title = "Troisi{\`e}me groupe de cohomologie non ramifi{\'e}e d'un solide cubique sur un corps de fonctions d'une variable",

abstract = "We prove that the third unramified cohomology group of a smooth cubic threefold over the function field of a complex curve vanishes. For this, we combine a method of C. Voisin with Galois descent on the codimension 2 Chow group. As a corollary, we show that the integral Hodge conjecture holds for degree 4 classes on smooth projective fourfolds equipped with a fibration over a curve, the generic fibre of which is a smooth cubic threefold, with arbitrary singularities on the special fibres.",

keywords = "Chow groups, Codimension 2 cycles, Family of cubic hypersurfaces, Integral Hodge conjecture, Intermediate jacobian, Unramified cohomology",

author = "Colliot-Th{\'e}l{\`e}ne, {Jean Louis} and Alena Pirutka",

note = "Funding Information: Remerciements. Ce travail a commenc{\'e} {\`a} Vienne pendant le programme “Advances in Birational Geometry 2017{"} ; nous remercions l{\textquoteright}Institut Schr{\"o}dinger pour son hospitalit{\'e} et Ludmil Katzarkov pour avoir lanc{\'e} ce programme. Le deuxi{\`e}me auteur remercie le Laboratoire de Sym{\'e}trie Miroir NRU HSE, RF Government grant, ag. 14.641.31.0001 et le NSF grant 1601680 pour leurs soutiens financiers. Le premier auteur remercie l{\textquoteright}Institut Courant, NYU, pour son hospitalit{\'e}. Publisher Copyright: {\textcopyright} by the author(s) Copyright: Copyright 2019 Elsevier B.V., All rights reserved.",

year = "2018",

language = "French",

volume = "2",

journal = "Epijournal de Geometrie Algebrique",

issn = "2491-6765",

publisher = "Association de l'Epijournal de Geometrie Algebrique",

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T1 - Troisième groupe de cohomologie non ramifiée d'un solide cubique sur un corps de fonctions d'une variable

AU - Colliot-Thélène, Jean Louis

AU - Pirutka, Alena

N1 - Funding Information: Remerciements. Ce travail a commencé à Vienne pendant le programme “Advances in Birational Geometry 2017" ; nous remercions l’Institut Schrödinger pour son hospitalité et Ludmil Katzarkov pour avoir lancé ce programme. Le deuxième auteur remercie le Laboratoire de Symétrie Miroir NRU HSE, RF Government grant, ag. 14.641.31.0001 et le NSF grant 1601680 pour leurs soutiens financiers. Le premier auteur remercie l’Institut Courant, NYU, pour son hospitalité. Publisher Copyright: © by the author(s) Copyright: Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2018

Y1 - 2018

N2 - We prove that the third unramified cohomology group of a smooth cubic threefold over the function field of a complex curve vanishes. For this, we combine a method of C. Voisin with Galois descent on the codimension 2 Chow group. As a corollary, we show that the integral Hodge conjecture holds for degree 4 classes on smooth projective fourfolds equipped with a fibration over a curve, the generic fibre of which is a smooth cubic threefold, with arbitrary singularities on the special fibres.

AB - We prove that the third unramified cohomology group of a smooth cubic threefold over the function field of a complex curve vanishes. For this, we combine a method of C. Voisin with Galois descent on the codimension 2 Chow group. As a corollary, we show that the integral Hodge conjecture holds for degree 4 classes on smooth projective fourfolds equipped with a fibration over a curve, the generic fibre of which is a smooth cubic threefold, with arbitrary singularities on the special fibres.

KW - Chow groups

KW - Codimension 2 cycles

KW - Family of cubic hypersurfaces

KW - Integral Hodge conjecture

KW - Intermediate jacobian

KW - Unramified cohomology

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AN - SCOPUS:85069558584

SN - 2491-6765

VL - 2

JO - Epijournal de Geometrie Algebrique

JF - Epijournal de Geometrie Algebrique

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ER -

Troisième groupe de cohomologie non ramifiée d'un solide cubique sur un corps de fonctions d'une variable

Abstract

Keywords

ASJC Scopus subject areas

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