## 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].

### MSC:

 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

### Keywords:

K-theory; Hodge theory; algebraic cycles

### Citations:

Zbl 0922.19002; Zbl 1273.14013
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