On complete intersections in varieties with finite-dimensional motive

TL;DR

This paper explores how the properties of a complete intersection within a variety with finite-dimensional motive influence the structure of Chow groups, extending previous results and providing new examples with finite-dimensional motives.

Contribution

It generalizes Voisin's result by linking niveau filtrations on homology and Chow groups, and introduces variants with group actions to produce new examples.

Findings

Conditions on niveau filtration affect Chow groups.

Generalization of Voisin's injectivity result.

New examples of varieties with finite-dimensional Chow motives.

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.