2-cycles sur les hypersurfaces cubiques de dimension 5

TL;DR

This paper investigates algebraic 2-cycles on smooth cubic hypersurfaces of dimension 5, proving the Griffiths group is trivial and that the Abel-Jacobi map is an isomorphism between certain Chow groups and the intermediate Jacobian.

Contribution

It establishes the triviality of the Griffiths group and the isomorphism induced by the Abel-Jacobi map for these hypersurfaces, advancing understanding of their algebraic cycles.

Findings

Griffiths group of 2-cycles is trivial

Abel-Jacobi map is an isomorphism for algebraically trivial 2-cycles

Chow group of 2-cycles is isomorphic to the intermediate Jacobian

Abstract

On \'etudie les cycles alg\'ebriques de codimension 3 sur les hypersurfaces cubiques lisses de dimension 5. Pour une telle hypersurface, on d\'emontre d'une part que son groupe de Griffiths des cycles de codimension 3 est trivial et d'autre part que l'application d'Abel-Jacobi induit un isomorphisme entre son groupe de Chow des cycles de codimension 3 alg\'ebriquement equivalents \`a z\'ero et sa jacobienne interm\'ediaire. ---------- We study 2-cycles of a smooth cubic hypersurface of dimension 5. We show that the Griffiths group of 2-cycles is trivial and the Abel-Jacobi map induces an isomorphism between the Chow group of algebraically trivial 2-cycles and the intermediate Jacobian.