Hypersurfaces quartiques de dimension 3 : non rationalité stable

Abstract

Inspirés par un argument de C. Voisin, nous montrons l'existence d'hypersurfaces quartiques lisses dans ${\mathbf{P}}_{\mathbb{C}}^4$ qui ne sont pas stablement rationnelles, plus précisément dont le groupe de Chow de degré zéro n'est pas universellement égal à $\mathbb{Z}$. --- There are (many) smooth quartic hypersurfaces in ${\mathbf{P}}_{\mathbb{C}}^4$ which are not stably rational. More precisely, their degree zero Chow group is not universally equal to $\mathbb{Z}$. The proof uses a variation of a specialisation method due to C. Voisin.

arXiv commentin French. More applications of the method are given : quartic threefolds defined over the algebraic closure of the rationals, cubic threefolds over a p-adic field. Final version, to appear in Annales Sc. Ec. Norm. Sup