Algebraic cycles and realification of complex projective varieties
For a real projective algebraic variety \(V \subset \mathbb{R} P^n\) one can define the group \(H^{2k}_{\text{alg}} (V, \mathbb{Z})\) for any nonnegative integer \(k\). One can also define the group \(H^{2k}_{\mathbb{C} \text{-alg}} (V, \mathbb{Z})\) as the group \(j^* (H^{2k}_{\text{alg}} (W, \mathbb{Z}) )\), where \(j : V \to W\) is the inclusion map and \(W \subset \mathbb{C} P^n\) is a complexification of \(V\). The author compares the group \(H^{2k}_{\text{alg}} (V, \mathbb{Z})\) and \(H^{2k}_{\mathbb{C} \text{-alg}} (V, \mathbb{Z})\) in the case where \(V\) is a complex projective variety equipped with its underlying real algebraic structure. First of all the author proves that \(H^{2k}_{\text{alg}} (V, \mathbb{Z})\) is a subgroup of \(H^{2k}_{\mathbb{C} \text{-alg}} (V, \mathbb{Z})\). So one can consider the quotient group \(\Gamma^{2k} (V)\). If \(f : V \to A\) is the Albanese map and \(V\) is nonsingular as a complex projective variety then it is shown that the induced map: \(f^\# : \Gamma^2 (A) \to \Gamma^2 (V)\) is surjective. The group \(\Gamma^2 (A)\) is computed explicitly for the case \(A = {\mathbb{C}^n/ L}\), where \(L\) is a lattice.
complex projective variety equipped with its underlying real algebraic structure
scientific article; zbMATH DE number 806216
scientific article