Real algebraic cycles on complex projective varieties
Let \(X\) be a nonsingular projective complex algebraic variety of dimension \(d\). We study the subgroup \(H_{2d-1}^{\text{alg}} (X_\mathbb{R},\mathbb{Z}/2\mathbb{Z})\) of homology classes represented by codimention 1 Zariski-closed subsets of the underlying real algebraic structure \(X_\mathbb{R}\) of \(X\). We prove, that if the Albanese variety of \(X\), denoted by \(\text{Alb }X\), is of dimension \(n\), then \(H_{2d-1}^{\text{alg}} (X_\mathbb{R},\mathbb{Z}/2\mathbb{Z})\) is canonically isomorphic to \(H_{2d- 1}^{\text{alg}}((\text{Alb } X)_\mathbb{R},\mathbb{Z}/2\mathbb{Z})\) and we see how this last group can be computed explicitly from a period matrix of \(\text{Alb }X\). Several applications are given.
fundamental class of real algebraic subset
scientific article; zbMATH DE number 1033928