On the Deligne-Beilinson cohomology sheaves. (English) Zbl 1346.14024

Let \(X\) be a compact algebraic manifold over \(\mathbb{C}\). The Deligne-Beilinson cohomology \(H^*(X,\mathbb{Z}(\cdot)_{{\mathcal D}})\) are defined by taking the hypercohomology of the truncated de Rham complex augmented over \(\mathbb{Z}\). The associated Deligne-Beilinson cohomology sheaves \({\mathcal H}^p_{{\mathcal D}}(\mathbb{Z})\), where \(\mathbb{Z}(r)(2\pi i)^r\mathbb{Z}\subset\mathbb{C}\), have a Gersten resolution \[ 0\to{\mathcal H}^o_{{\mathcal D}}\Biggl(\mathbb{Z}\to \coprod_{s\in X^0} (i_x)* H^p_{{\mathcal D}}(\mathbb{C}(x))\Biggr),\;\mathbb{Z}\to \coprod_{x\in X^1} (i_x)* H^{p-1}_{{\mathcal D}}(\mathbb{C}(x),\mathbb{Z}(r-1))\to\cdots \] The above revolution yields an isomorphism of graded rings \[ \eta:\bigoplus CH^p(X)\simeq \bigoplus H^p(X,{\mathcal H}^p_{{\mathcal D}}(\mathbb{Z}(p))), \] where the intersection product of algebraic cycles correspond to the cup product in Deligne-Beilinson cohomology.
In this paper, the author shows that the Bloch-Kato conjecture, as proved by Voevodsky and Rost, implies that \({\mathcal H}^{q+1}_{{\mathcal D}}(\mathbb{Z}(q))\) is torsion-free. Therefore \(H^0(X,{\mathcal H}^{q+1}_{{\mathcal D}}(\mathbb{Z}(q)))= 0\) if \(X\) is unirational. In the case \(S\) is a smooth projective surface over \(\mathbb{C}\), then the group \(H^0(S,{\mathcal H}^2_{{\mathcal D}}(\mathbb{Z}(2)))\) is uniquely divisible.
If moreover \(p_g(S)=0\), then \(H^0(S,{\mathcal H}^3_{{\mathcal D}}(\mathbb{Z}(2)))= T(X)\), where \(T(X)\) is the Albanese kernel, a result that already appeared by A. Rosenschon [\(K\)-Theory 16, No. 2, 185–199 (1999; Zbl 0922.19002), Theorem 6.1)] and by the reviewer [Clay Math. Proc. 9, 53–74 (2010; Zbl 1273.14013), Theorem 6.3].


14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F42 Motivic cohomology; motivic homotopy theory
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