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arXiv · 1709.00597

Troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique sur un corps de fonctions d'une variable

Jean-Louis Colliot-Thélène·Alena Pirutka
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Abstract

En combinant une m\'ethode de C. Voisin avec la descente galoisienne sur le groupe de Chow en codimension $2$, nous montrons que le troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique lisse d\'efini sur le corps des fonctions d'une courbe complexe est nul. Ceci implique que la conjecture de Hodge enti\`ere pour les classes de degr\'e 4 vaut pour les vari\'et\'es projectives et lisses de dimension 4 fibr\'ees en solides cubiques au-dessus d'une courbe, sans restriction sur les fibres singuli\`eres. --------------- 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.

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Jean-Louis Colliot-Thélène, Alena Pirutka. 2017-09-02. Troisi\`eme groupe de cohomologie non ramifi\'ee d'un solide cubique sur un corps de fonctions d'une variable. https://doi.org/10.46298/epiga.2018.volume2.3950

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