Finite automorphism groups of complex tori of dimension two
type: Item
instance of: scholarly article
title: Finite automorphism groups of complex tori of dimension two
zbMATH Open document ID: 0654.32015
author: Akira Fujiki
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1988-01-01T00:00:00Z
review text: Let (T,G) be a pair consisting of a (possibly non-algebraic) complex torus T and a finite subgroup G of Aut(T), where Aut is the automorphism group as a complex Lie group. This paper gives a complete classification of such pairs with dim T\(=2\) up to isomorphisms. In particular G is considered modulo conjugacy in Aut(T). The results are very precise and summarized in many classification tables. Here only a rough outline of them will be sketched and a few sample results will be mentioned. The actions of G on \(H_ 1(T;{\mathbb{Z}})\) and H 0(T,\(\Theta)\) give two representations into \(GL_ 4({\mathbb{Z}})\) and \(GL_ 2({\mathbb{C}})\). They are called rational and complex representation types of (T,G) respectively. The strategy is first to classify all the possible compatible rational and complex representation types, next to determine the structure of the moduli space \({\mathcal M}\) consisting of isomorphism classes of (T,G) with fixed rational and complex representation types, and then to describe the structure of each member of M as explicitly as possible (\({\mathcal M}\) is a quotient of a homogeneous manifold by a discrete group, but the action may be bad and \({\mathcal M}\) may have no analytic structure). The author studies and classifies also rational endomorphism rings of T. The pair (T,G) is called special if the complex representation factors through \(SL_ 2({\mathbb{C}})\). Such pairs are studied more in detail. The lattice structure of H 2(T;\({\mathbb{Z}})\) is studied and several interesting relations are given. For example, a necessary and sufficient condition for a singular abelian surface T to admit a special action of a given group G is given in terms of the Neron-Severi lattice of T.
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 22E10
Mathematics Subject Classification ID: 32G13
zbMATH DE Number: 4067337
zbMATH Keywords: complex torus
zbMATH Keywords: automorphism group
zbMATH Keywords: representations
reviewed by: Takao Fujita
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195175326
OpenAlex ID: W2062410813
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DOI: 10.2977/PRIMS/1195175326
description: scientific article; zbMATH DE number 4067337