On complete intersections in varieties with finite-dimensional motive

Abstract

Let X be a complete intersection inside a variety M with finite-dimensional motive and for which the Lefschetz-type conjecture $B(M)$ holds. We show how conditions on the niveau filtration on the homology of X influence directly the niveau on the level of Chow groups. This leads to a generalization of Voisin’s result. The latter states that if M has trivial Chow groups and if X has non-trivial variable cohomology parametrized by c-dimensional algebraic cycles, then the cycle class maps$A_k(X)\to H_{2k}(X)$are injective for$k<c$⁠. We give variants involving group actions, which lead to several new examples with finite-dimensional Chow motives.