Algebraic cycles and realification of complex projective varieties
type: Item
instance of: scholarly article
title: Algebraic cycles and realification of complex projective varieties
zbMATH Open document ID: 0839.14010
DOI: 10.1007/BF01265347
published in: Geometriae Dedicata
publication date: 1996-06-23T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14F25
Mathematics Subject Classification ID: 14P05
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 806216
zbMATH Keywords: algebraic cycles
zbMATH Keywords: cohomology group
zbMATH Keywords: complex projective variety equipped with its underlying real algebraic structure
author: Wojciech Kucharz
reviewed by: Fabrizio Broglia
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 806216