Quelques cas d'annulation du troisi\`eme groupe de cohomologie non ramifi\'ee

TL;DR

This paper proves the vanishing of the third unramified cohomology group for certain varieties with specific geometric structures, leading to confirmation of the integral Hodge conjecture in degree 4 over complex fields.

Contribution

It establishes the nullity of the third unramified cohomology group for varieties with quadrics or complete intersections of two quadrics, advancing understanding of the integral Hodge conjecture.

Findings

Vanishing of the third unramified cohomology group for specified varieties.

Confirmation of the integral Hodge conjecture in degree 4 over complex fields.

Extension of cohomological results to geometrically significant classes of varieties.

Abstract

The third unramified cohomology group is shown to vanish on certain varieties equipped with a pencil of quadrics or of smooth complete intersections of two quadrics. Over the complex field, this shows that the integral Hodge conjecture in degree 4 holds for such varieties. --- On \'etablit la nullit\'e du troisi\`eme groupe de cohomologie non ramifi\'ee pour certaines vari\'et\'es munies d'un pinceau de quadriques ou d'intersections compl\`etes lisses de deux quadriques. Sur les complexes, ceci permet d'\'etablir la conjecture de Hodge enti\`ere en degr\'e 4 pour de telles vari\'et\'es.