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Zero-cycles and cohomology on real algebraic varieties. (English) Zbl 0882.14016
Summary: Let $$X$$ be an algebraic variety over $$\mathbb{R}$$, the field of real numbers. The interplay between the topology of the set of real points $$X(\mathbb{R})$$ and the algebraic geometry of $$X$$ has been the object of much study (Harnack, Weichold, Witt, Geyer, Artin/Verdier and Cox). In the present paper, we first analyze the Chow group $$\text{CH}_0 (X)$$ of zero-cycles on $$X$$ modulo rational equivalence. Let $$t$$ be the number of compact connected components of $$X(\mathbb{R})$$. The quotient of $$\text{CH}_0 (X)$$ by its maximal divisible subgroup is a finite group, equal to $$(\mathbb{Z}/2)^t$$ if $$X (\mathbb{R}) \neq \emptyset$$. For $$X/ \mathbb{R}$$ smooth and proper we compute the torsion subgroup of $$\text{CH}_0 (X)$$ (we use Roitman’s theorem over $$\mathbb{C})$$.
Let $$X/ \mathbb{R}$$ be smooth, connected, $$d$$-dimensional and assume $$X(\mathbb{R}) \neq \emptyset$$. We use the Artin-Verdier-Cox results to analyze the Bloch-Ogus spectral sequence $$E_2^{pq} =H^p_{\text{Zar}} (X,{\mathcal H}^q) \Rightarrow H^{p+q}_{\text{et}} (X, \mathbb{Z}/ 2)$$. Here the Zariski sheaves $${\mathcal H}^q$$ are the sheaves obtained by sheafifying étale cohomology (with coefficients $$\mathbb{Z}/2)$$. We show that in degrees high enough this spectral sequence degenerates and that many groups $$H^p_{\text{Zar}} (X, {\mathcal H}^q)$$ are finite. A new proof of the isomorphism $$\text{CH}_0 (X)/2 \cong (\mathbb{Z}/2)^t$$ is given, and the cycle map $$\text{CH}_0 (X)/2 \to H^{2d}_{\text{et}} (X,\mathbb{Z}/2)$$ is shown to be injective. The group $$H^{d-1} (X(\mathbb{R}), \mathbb{Z}/2)$$ is shown to be a quotient of $$H^{d-1} (X, {\mathcal H}^d)$$. If $$H^{2d-1} (X_\mathbb{C}, \mathbb{Z}/2)=0$$, then $$H^{d-2} (X(\mathbb{R}), \mathbb{Z}/2)$$ is a quotient of $$H^{d-2} (X, {\mathcal H}^d)$$. There is a natural map $$H^{d-1} (X, {\mathcal K}_d)/2 \to H^{d-1} (X(\mathbb{R}), \mathbb{Z}/2)$$. Sufficient conditions for it to be an isomorphism are given (e.g. $$X_\mathbb{C}$$ projective and simply connected).

##### MSC:
 14P25 Topology of real algebraic varieties 14F25 Classical real and complex (co)homology in algebraic geometry 14C15 (Equivariant) Chow groups and rings; motives 14C05 Parametrization (Chow and Hilbert schemes)
##### Keywords:
real points; Chow group; spectral sequence; cycle map
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