On the Deligne-Beilinson cohomology sheaves

We prove that the Deligne–Beilinson cohomology sheaves Hq+1(Z(q)D) are torsion-free as a consequence of the Bloch–Kato conjectures as proven by Rost and Voevodsky. This implies that H0(X,Hq+1(Z(q)D))=0 if X is unirational. For a surface X with pg=0 we show that the Albanese kernel, identified with H0(X,H3(Z(2)D)), can be characterized using the integral part of the sheaves associated to the Hodge filtration.