Deligne–Beilinson cycle maps for Lichtenbaum cohomology

We define Deligne–Beilinson cycle maps for Lichtenbaum cohomology of arbitrary complex algebraic varieties and show that the analogues of the Abel–Jacobi theorem and the Lefschetz theorem on $(1, 1)$-cycles hold for any complex algebraic variety if we replace the divisor class group with Voevodsky’s motivic cohomology with compact supports. For more general indices, we study the torsion part of the cycle maps. We also characterize the algebraic part of Griffiths’s intermediate Jacobians by a universal property.

Keywords

étale motivic cohomology, Deligne cohomology, cycle map

2010 Mathematics Subject Classification

14C30, 14F42, 19E15

Full Text (PDF format)

Received 5 October 2017

Received revised 20 June 2018

Published 10 October 2018