$CH_{0}$-trivialit\'e universelle d'hypersurfaces cubiques presque diagonales

TL;DR

This paper proves that certain smooth cubic hypersurfaces, defined by sums of small-variable cubic forms, are universally $CH_{0}$-trivial, implying they have a specific algebraic decomposition property.

Contribution

It establishes the universal $CH_{0}$-triviality for a new class of cubic hypersurfaces characterized by their defining equations involving limited-variable cubic forms.

Findings

Hypersurfaces with equations as sums of at most three-variable cubic forms are universally $CH_{0}$-trivial.

The result applies to all smooth cubic hypersurfaces of dimension at least 2 fitting this form.

The proof involves algebraic and geometric techniques related to Chow groups and decompositions of the diagonal.

Abstract

Toute hypersurface cubique lisse complexe de dimension au moins 2 dont l'\'equation est donn\'ee par l'annulation d'une somme de formes cubiques \`a variables s\'epar\'ees, chaque forme impliquant au plus trois variables, est universellement $CH_{0}$-triviale. --- If a smooth cubic hypersurface of dimension at least 2 is defined by the vanishing of a sum of cubic forms in independent variables and each of these forms involves at most 3 variables, then the cubic hypersurface is universally $CH_{0}$-trivial : there is an integral Chow decomposition of the diagonal.