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 CH,(X) of zero-cycles on X module 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 (Z/2)(t) if X(R) not equal 0. For X/R smooth and proper we compute the torsion subgroup of CH0(X) (we use Roitman's theorem over C). Let X/R be smooth, connected, d-dimensional and assume X(R) not equal 0. We use the Artin/Verdier/Cox results to analyze the Bloch-Ogus spectral sequence E(2)(pq) = H-2ar(p)(X, H-q), H-et(p+q)(X, Z/2). Here the Zariski sheaves H-q are the sheaves obtained by sheafifying etale cohomology (with coefficients Z/2). We show that in high enough degrees this spectral sequence degenerates and that many groups H-2ar(p)(X, H-q) are finite. A new proof of the isomorphism CH0(X)2 congruent to (Z/2)(t) is given, and the cycle map CH0(X)/2 --> H-et(2P)(X, Z/2) is shown to be injective. The group H-d-1(X(R), Z/2) is shown to be a quotient of H-d-1(X, H-d). If H-2d-1(X(C), Z/2) = 0, then H-d-2(X(R), Z/2) is a quotient of H-d-2(X, H-d). There is a natural map H-d-1(X, H-d)/2 --> H-d-1(X(R), Z/2). Sufficient conditions for it to be an isomorphism are given (e.g. X(c) projective and simply connected).