Finite automorphism groups of complex tori of dimension two. (English) Zbl 0654.32015

Fujiki, Akira

Publ. Res. Inst. Math. Sci. 24, No. 1, 1-97 (1988).

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.

Reviewer: T.Fujita

MSC:

32J15	Compact complex surfaces
32M05	Complex Lie groups, group actions on complex spaces
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
22E10	General properties and structure of complex Lie groups
32G13	Complex-analytic moduli problems

Keywords:

complex torus; automorphism group; representations

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