Real algebraic cycles on complex projective varieties

Abstract

If we identify CN with R2N , then to any (quasi-)affine complex algebraic variety V of dimension d corresponds in a natural way a real algebraic variety of dimension 2d , in the sense of [BCR], which we will denote by VR. Since each complex morphism V → V ′ can be considered as a real morphism VR → V ′ R, and all complex algebraic varieties are obtained by glueing affine pieces together, we can extend this construction in a natural way to arbitrary complex algebraic varieties X in order to obtain XR, the underlying real algebraic structure of X . If f : X → X ′ is a morphism of complex algebraic varieties, then it is also in a natural way a regular mapping of real structures XR → X ′ R, which, by abuse of notation, will be denoted by f as well. If Y is a real algebraic variety of dimension n and Hk (Y ,Z/2Z) is the singular homology group in dimension k (with respect to the strong topology), then for all k ≤ n there is a subgroup, denoted by H alg k (Y ,Z/2Z), consisting of the fundamental classes of k-dimensional real algebraic subsets of Y . See [BCR] for a definition of the fundamental class of a real algebraic set and [BK1] for more information on H alg k . We will study the group H alg 2d−1(Y ,Z/2Z) for Y = XR, where X is a nonsingular projective complex algebraic variety of dimension d . In particular, Y is a nonsingular real algebraic variety of dimension n = 2d that is compact for the strong topology. It will be more convenient to consider H 1 alg(Y ,Z/2Z), which is by definition the image of H alg n−1(Y ,Z/2Z) under the Poincare duality isomorphism Hn−1(Y ,Z/2Z) → H 1(Y ,Z/2Z). Then the homomorphism