The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

TL;DR

This paper demonstrates that, assuming the Lefschetz standard conjecture, the triviality of Chow groups for certain complete intersections is equivalent to a geometric coniveau condition on their vanishing cohomology, linking algebraic cycles and Hodge theory.

Contribution

It establishes the equivalence between the generalized Hodge and Bloch conjectures for general complete intersections under the Lefschetz standard conjecture.

Findings

Chow groups of certain cycles are trivial under specific coniveau conditions.

The equivalence of generalized Hodge and Bloch conjectures is proven for general complete intersections.

Assumption of the Lefschetz standard conjecture is crucial for the results.

Abstract

Let $X$ be a smooth complex projective variety with trivial Chow groups. (By trivial, we mean that the cycle class is injective.) We show (assuming the Lefschetz standard conjecture) that if the vanishing cohomology of a general complete intersection $Y$ of ample hypersurfaces in $X$ has geometric coniveau $\geq c$, then the Chow groups of cycles of dimension $\leq c-1$ of $Y$ are trivial. The generalized Bloch conjecture for $Y$ is this statement with "geometric coniveau" replaced by "Hodge coniveau".