Real algebraic cycles on complex projective varieties
type: Item
instance of: scholarly article
title: Real algebraic cycles on complex projective varieties
zbMATH Open document ID: 0874.14005
DOI: 10.1007/PL00004309
author: J. van Hamel
published in: Mathematische Zeitschrift
publication date: 1997-07-15T00:00:00Z
review text: 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.
reviewed by: J. van Hamel
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 32G20
zbMATH DE Number: 1033928
zbMATH Keywords: fundamental class of real algebraic subset
zbMATH Keywords: complex algebraic variety
zbMATH Keywords: real algebraic structure
zbMATH Keywords: Albanese variety
zbMATH Keywords: period matrix
MaRDI profile type: Publication
full work available at URL: pl00004309
OpenAlex ID: W1984776648
description: scientific article; zbMATH DE number 1033928