On cycles on compact locally symmetric varieties
type: Item
instance of: scholarly article
title: On cycles on compact locally symmetric varieties
zbMATH Open document ID: 1026.11052
published in: Monatshefte für Mathematik
publication date: 2002-06-27T00:00:00Z
review text: Let \(G\) be a connected simply connected semisimple algebraic group defined and anisotropic over \(\mathbb Q\), and let \(\Gamma\) be a torsion-free cocompact arithmetic subgroup of \(G(\mathbb Q)\). Assuming that the Riemannian symmetric space \(X = G(\mathbb R)/K_\infty\) with \(K_\infty\) being a maximal compact subgroup has a \(G(\mathbb R)\)-invariant complex structure, it is well-known that the locally symmetric space \(S(\Gamma) = \Gamma \backslash X\) has the structure of a smooth projective variety. In this paper the author obtains a criterion to determine when the cycle class of a locally symmetric subvariety \(S_H(\Gamma)\) of \(S(\Gamma)\) generates a nontrivial module under the action of Hecke operators and provide examples of such subvarieties. He also proves that all Hodge classes in degree \(4n-4\) on \(S(\Gamma)\) associated to certain arithmetic subgroups \(\Gamma\) of \(\text{SU}(2,n)\) are algebraic provided that \(n \geq 5\).
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 1761124
zbMATH Keywords: locally symmetric varieties
zbMATH Keywords: Shimura varieties
zbMATH Keywords: Hodge classes
zbMATH Keywords: Hecke operators
author: T. N. Venkataramana
MaRDI profile type: Publication
full work available at URL: s006050200018
OpenAlex ID: W2018481429
DOI: 10.1007/S006050200018
description: scientific article; zbMATH DE number 1761124