## Abel-Jacobi Equivalence and a Variant of the Beilinson-Hodge Conjecture

J. Math. Sci. Univ. Tokyo
Vol. 17 (2010), No. 2, Page 179--199.

Lewis, James D.
Abel-Jacobi Equivalence and a Variant of the Beilinson-Hodge Conjecture
Let $X/\C$ be a smooth projective variety and $\Ch^r(X)$ the Chow group of codimension $r$ algebraic cycles modulo rational equivalence. Let us assume the (conjectured) existence of the Bloch-Beilinson filtration $\{F^{\nu}\Ch^r(X)\otimes\Q\}_{\nu=0}^r$ for all such $X$ (and $r$). If $\Ch^r_{AJ}(X)\subset \Ch^r(X)$ is the subgroup of cycles Abel-Jacobi equivalent to zero, then there is an inclusion $F^2\Ch^r(X)\otimes\Q \subset \Ch^r_{AJ}(X)\otimes\Q$. Roughly speaking we show that this inclusion is an equality for all $X$ (and $r$) if and only if a certain variant of Beilinson-Hodge conjecture holds for $K_1$.