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Beilinson-Hodge cycles on semiabelian varieties. (English) Zbl 1182.14005
Let $$U/{\mathbb C}$$ be a smooth complex quasiprojective variety, and $$CH^i(U,j)$$ the higher Chow group introduced by S. Bloch. Concerning the cycle class map $cl_{i,j} : CH^i(U,j)\otimes{\mathbb Q} \to \hom_{\text{MHS}}({\mathbb Q}(0),H^{2i-j}(U,{\mathbb Q}(i))),$ Beilinson conjectured that $$cl_{i,j}$$ is surjective, where the case $$j=0$$ amounts to the statement of the Hodge conjecture. It was Jannsen who first found a counterexample in the case $$i > j=1$$, and since then other counterexamples where found by the reviewer and M. Kerr [Invent. Math. 170, No. 2, 355–420 (2007; Zbl 1139.14010)]. Anecdotal evidence based on examples in the works of M. Asakura and S. Saito suggests that this conjecture holds in the case $$i=j$$, which is now coined the Beilinson-Hodge conjecture. In the paper under review, the authors give an interesting proof of the Beilinson-Hodge conjecture for $$U$$ being a product of curves or a semiabelian variety. Their method is based on a study of invariants under the Mumford-Tate group, reducing to the situation where $$i=j=1$$, where the conjecture is easily known to hold.

##### MSC:
 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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