Beilinson-Tate cycles on semiabelian varieties

TL;DR

This paper proves Beilinson-Hodge and Beilinson-Tate conjectures for varieties dominated by products of curves, establishing that certain Hodge and Tate cycles originate from higher Chow groups and are algebraic.

Contribution

The authors extend previous results by proving Beilinson-Hodge and Beilinson-Tate conjectures for a broader class of varieties dominated by products of curves.

Findings

Beilinson-Hodge conjecture verified for varieties dominated by products of curves.

Beilinson-Tate conjecture verified for varieties dominated by products of curves.

Lower weight Hodge and Tate cycles are shown to be algebraic in these cases.

Abstract

Along the lines of Hodge and Tate conjectures, Beilinson conjectured that in the qth cohomology all the weight 2q Hodge cycles of a smooth complex variety and all the weight 2q Tate cycles of a smooth variety over a finitely generated field comes from the higher Chow groups. For product of curves and semiabelian varieties, Beilinson-Hodge conjecture was shown in a previous paper by the authors. Here both Beilinson-Hodge and Beilinson-Tate conjectures are shown to be true for varieties dominated by product of curves. We also show that lower weight Hodge cycles (resp. Tate cycles) are algebraic in these situations.