ARITHMETIC NORMAL FUNCTIONS AND FILTRATIONS ON CHOW GROUPS

Let X/C be a smooth projective variety, and let CHr(X, m) be the higher Chow group defined by Bloch. Saito and Asakura defined a descending candidate Bloch-Beilinson filtration CHr (X, m; Q) = F-0 superset of ... superset of F-r superset of Fr+1 = Fr+2 = ..., using the language of mixed Hodge modules. Another more geometrically defined filtration is constructed by Kerr and Lewis in terms of germs of normal functions. We show that under the assumptions (i) X/C = X-0 x C where X-0 is defined over (Q) over bar, and (ii) the general Hodge conjecture, that (FCHr)-C-center dot(X, m; Q) coincides with the aforementioned geometric filtration. More specifically, it is characterized in terms of germs of reduced arithmetic normal functions.