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

TL;DR

This paper proves the vanishing of the third unramified cohomology group for smooth cubic threefolds over function fields of complex curves, leading to new cases of the integral Hodge conjecture in fourfolds.

Contribution

It introduces a novel combination of Voisin's method with Galois descent to establish the vanishing of a key cohomology group for cubic threefolds over function fields.

Findings

Third unramified cohomology group of smooth cubic threefolds over complex function fields is zero.

The integral Hodge conjecture holds for degree 4 classes on certain fourfolds with cubic threefold fibrations.

Results apply even with singular fibers in the fibration.

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.