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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AU - Colliot-Thélène, Jean Louis

AU - Pirutka, Alena

N1 - Publisher Copyright: © by the author(s)

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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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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