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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).

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)
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