Étale monodromy and rational equivalence for -cycles on cubic hypersurfaces in

Abstract

Abstract Let be an uncountable algebraically closed field of characteristic , and let be a smooth projective connected variety of dimension , embedded into over . Let be a hyperplane section of , and let and be the groups of algebraically trivial algebraic cycles of codimension and modulo rational equivalence on and , respectively. Assume that, whenever is smooth, the group is regularly parametrized by an abelian variety and coincides with the subgroup of degree classes in the Chow group . We prove that the kernel of the push-forward homomorphism from to is the union of a countable collection of shifts of a certain abelian subvariety inside . For a very general hyperplane section either or coincides with an abelian subvariety in whose tangent space is the group of vanishing cycles . Then we apply these general results to sections of a smooth cubic fourfold in . Bibliography: 33 titles.