For a smooth complex projective variety X deﬁned over a number ﬁeld, we have ﬁltrations on the Chow groups depending on the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained ﬁltration coincides with the ﬁltration of Green and Griﬃths, assuming the Hodge conjecture. In the case the realizations contain Hodge structure and etale cohomology, we prove that if the second graded piece of the ﬁltration does not vanish, it contains a nonzero element which is represented by a cycle deﬁned over a ﬁeld of transcendence degree one. This may be viewed as a reﬁnement of results of Nori, Schoen, and Green–Griffiths–Paranjape. For higher graded pieces we have a similar assertion assuming a conjecture of Beilinson and Grothendieck’s generalized Hodge conjecture.
Cite this article
Morihiko Saito, Filtrations on Chow Groups and Transcendence Degree. Publ. Res. Inst. Math. Sci. 40 (2004), no. 3, pp. 933–948DOI 10.2977/PRIMS/1145475497