Endomorphism algebras of complex tori
type: Item
instance of: scholarly article
title: Endomorphism algebras of complex tori
zbMATH Open document ID: 0858.14024
DOI: 10.1007/BF01460976
author: Frans Oort
author: Yu. G. Zarhin
published in: Mathematische Annalen
publication date: 1997-03-25T00:00:00Z
full work available at URL: https://eudml.org/doc/165354
review text: It is known that not every finite-dimensional associative algebra \(D\) over \(\mathbb{Q}\) can be realized as the algebra of endomorphisms of an abelian variety tensored with \(\mathbb{Q}\). The necessary and sufficient condition is the existence of a positive definite anti-involution on \(D\). The authors prove that for complex tori this is not true. Namely, for any \(D\) as above there exists a complex torus \(T\) such that \(\text{End} (T)\otimes \mathbb{Q} \cong D\). Moreover, if \(n= \text{rank}_\mathbb{Q} A\) then one may choose \(T\) of dimension \(t \cdot n\), where \(t\) is any positive integer. Also \(T\) is semisimple if and only if the algebra \(D\) is semisimple. The authors apply this result to give new examples of non-algebraizable complex tori.
Mathematics Subject Classification ID: 14K20
Mathematics Subject Classification ID: 32J99
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 14E99
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 840171
zbMATH Keywords: algebra of endomorphisms of an abelian variety
zbMATH Keywords: non-algebraizable complex tori
reviewed by: Igor V. Dolgachev
MaRDI profile type: Publication
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OpenAlex ID: W1987716390
description: scientific article; zbMATH DE number 840171