Real divisors algebraically equivalent to zero
Let~\(X\) be a nonsingular projective real algebraic variety. The author studies the subgroup \(\text{Alg}^1(X(\mathbb{R}))\) of \(H^1(X(\mathbb{R}), \mathbb{Z}/2\mathbb{Z})\) consisting of cohomology classes represented by codimension \(1\) cycles on~\(X\) that are algebraically equivalent to \(0\). He shows that this subgroup is equal to the natural image of the Galois cohomology group \(H^2(G,H^1(X(\mathbb{C}),\mathbb{Z}))\) in \(H^1(X(\mathbb{R}), \mathbb{Z}/2\mathbb{Z})\). The idea of proof is the following. Firstly, using the Albanese variety of a variety, one shows that it suffices to prove the statement for an abelian variety. Secondly, for an abelian variety, the statement is proved using ``classical results [\textit{A. Comessatti}, Annali di Mat. (4) 2, 67--106 (1924; JFM 50.0641.01)], whose modern treatment is contained in [\textit{C. Ciliberto} and \textit{C.Pedrini}, in: Lectures in real geometry. De Gruyter Expo. Math. 23, 167--256 (1996; Zbl 0895.14013), \textit{B. Gross} and \textit{J. Harris}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 14, 157--182 (1981; Zbl 0533.14011), \textit{J. Huisman}, Ph.D. thesis, Amsterdam 1992 and \textit{R. Silhol}, ``Real algebraic surfaces'', Lect. Notes Math. 1392 (1989; Zbl 0691.14010)].
scientific article; zbMATH DE number 2064735