Abel-Jacobi equivalence and a variant of the Beilinson-Hodge conjecture (Q2902431)

As constructed by author in [\textit{J. D. Lewis}, Compos. Math. 128, No. 3, 299--322 (2001; Zbl 1052.14015)], there is a descending filtration, denoted by \(F^\bullet\) on the Chow groups of smooth projective complex varieties, which satisfies all of the expected Bloch-Beilinson type properties (cf. Section 4), except that the desired finiteness condition is not known. We remark that the finiteness condition is a consequence of the Hodge conjecture and a conjecture independently formualted by Bloch and Beilinson which says that the Abel-Jacobi map is injective for Chow groups of smooth proper varieties defined over number field.NEWLINENEWLINEWe always have that cycles in \(F^2 \mathrm{CH}_{\mathbb Q}\) are in the kernel of the Abel-Jacobi map, the paper under review mainly concerns about the following question: do we have an equality \(F^2 \mathrm{CH}_{\mathbb Q}=\mathrm{CH}_{AJ,\mathbb Q}\)?NEWLINENEWLINEThe main result of the paper is the following: provided that the Hodge conjecture is true and that the limit of the aforementioned filtration is homologically trivial in codimension 1, then the above equality is equivalent to the Beilinson's Hodge conjecture (amended by \textit{S.-J. Kang} and \textit{J. D. Lewis} in [Studies in Mathematics. Tata Institute of Fundamental Research 21, 197--215 (2010; Zbl 1273.14021)]) which says that the cycle class map on the higher Chow group \(\mathrm{CH}^r(\mathbb C(X),1)_{\mathbb Q}\to \Hom_{MHS}({\mathbb Q}, H^{2r-1}({\mathbb C}(X), {\mathbb Q}(r)))\) is surjective. The paper also presents some unconditional results in lower dimension or specific geometric situation to support the two conjecturally equivalent statements in the main theorem.