Zero-cycles and cohomology on real algebraic varieties

Let X be an algebraic variety over R, the field of real numbers. The interplay between the topology of the set of real points X(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 CH0(X) of zero-cycles on X modulo rational equivalence. Let t be the number of compact connected components of X(R). The quotient of CH0(X) by its maximal divisible subgroup is a finite group, equal to (Z2)t if X(R) ≠ φ. For X/R smooth and proper we compute the torsion subgroup of CH0(X) (we use Roitmanʹs theorem over C). R be smooth, connected, d-dimensional and assume X(R) ≠ φ. We use the Artin/Verdier/Cox results to analyze the Bloch-Ogus spectral sequence E2pq = HZarp(X, Hq) ⇒ Hétp + q(X, Z2). Here the Zariski sheaves Hq are the sheaves obtained by sheafifying étale cohomology (with coefficients Z2). We show that in high enough degrees this spectral sequence degenerates and that many groups HZarp(X, Hq) are finite. A new proof of the isomorphism CH0(X)2 ≅ (Z2)t is given, and the cycle map CH0(X)2 → Hét2d (X, Z2) is shown to be injective. The group Hd − 1X(R), Z2) is shown to be a quotient of Hd − 1(X, Hd). If H2d − 1(Xc, Z2) = 0, then Hd − 2(X(R), Z2) is a quotient of Hd − 2(X,Hd). There is a natural map Hd − 1(X, Kd)2 → Hd − 1(X(R), Z2). Sufficient conditions for it to be an isomorphism are given (e.g. Xc projective and simply connected).