# Deligne-Beilinson cycle maps for Lichtenbaum cohomology

## TL;DR

This paper introduces Deligne-Beilinson cycle maps for Lichtenbaum cohomology on complex algebraic varieties, generalizing classical theorems and analyzing their torsion properties and relations to intermediate Jacobians.

## Contribution

It defines new cycle maps for Lichtenbaum cohomology, extending classical results like Abel-Jacobi and Lefschetz theorems, and characterizes their torsion behavior and algebraic parts of Griffiths's Jacobians.

## Key findings

- Cycle maps are surjective on torsion with torsion-free cokernels.
- When m ≤ 2n, the cycle map with compact supports is an isomorphism on torsion.
- Characterization of the algebraic part of Griffiths's intermediate Jacobians.

## 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 $m \leq 2n,$ 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.