Deligne-Beilinson cycle maps for Lichtenbaum cohomology

Abstract

We define Deligne-Beilinson cycle maps for Lichtenbaum cohomology $H_L^m(X,\mathbb{Z}(n))$ and that with compact supports $H_{c,L}^m(X,\mathbb{Z}(n))$ of an arbitrary complex algebraic variety $X.$ When $(m,n)=(2,1),$ the homological part of our cycle map with compact supports gives a generalization of the Abel-Jacobi theorem and its projection to the Betti cohomology yields that of the Lefschetz theorem on $(1,1)$-cycles for arbitrary complex algebraic varieties. In general degrees $(m,n),$ we show that the Deligne-Beilinson cycle maps are always surjective on torsion and have torsion-free cokernels. If $m\le 2n,$ the version with compact supports induces an isomorphism on torsion, and so does the one without compact supports if $min\{2m-1,2\dim X+1\}\le 2n.$ We also characterize the algebraic part of Griffiths's intermediate Jacobians with a universal property.