  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 $m
\leq 2n,$ the version with compact supports induces an isomorphism on torsion,
and so does the one without compact supports if $min \{2m-1, 2 \dim X+1 \} \leq
2n.$ We also characterize the algebraic part of Griffiths's intermediate
Jacobians with a universal property.
