On the Deligne–Beilinson cohomology sheaves

Abstract

We prove that the Deligne–Beilinson cohomology sheaves ${\mathcal{\mathscr{H}}}^{q+1}\left(\mathbb{Z}{\left(q\right)}_{\mathcal{D}}\right)$ are torsion-free as a consequence of the Bloch–Kato conjectures as proven by Rost and Voevodsky. This implies that $H^0\left(X,{\mathcal{\mathscr{H}}}^{q+1}\left(\mathbb{Z}{\left(q\right)}_{\mathcal{D}}\right)\right)=0$ if $X$ is unirational. For a surface $X$ with $p_g=0$ we show that the Albanese kernel, identified with $H^0\left(X,{\mathcal{\mathscr{H}}}^3\left(\mathbb{Z}{\left(2\right)}_{\mathcal{D}}\right)\right)$, can be characterized using the integral part of the sheaves associated to the Hodge filtration.