Résumé. -Inspirés par un argument de C. Voisin, nous montrons l'existence d'hypersurfaces quartiques lisses de dimension 3 sur les complexes qui ne sont pas stablement rationnelles, plus précisément dont le groupe de Chow de degré zéro n'est pas universellementégalà Z. La méthode de spécialisation adoptée ici permet de construire des exemples définis sur un corps de nombres.Abstract. -There are (many) smooth quartic threefolds over the complex field which are not stably rational. More precisely, their degree zero Chow group is not universally equal to Z. The proof uses a variation of a method due to C. Voisin. The specialisation argument we use yields examples defined over a number field.

We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers. GeneralitiesWe recall implications of the "integral decomposition of the diagonal and specialization" method, following [CTP14], [Voi15b], and [Pir16].A projective variety X over a field k is universally CH 0 -trivial if for all field extensions k ′ /k the natural degree homomorphism from the Chow group of zero-cycles CH 0 (X k ′ ) → Z is an isomorphism. Examples include smooth k-rational varieties. More complicated examples arise as follows:

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