Complement to explicit description of Hopf surfaces and their automorphism groups

Abstract

In the previous paper we determined Aut(X) of each Hopf surface X W G with W C 2 (0, 0) so that its holomorphic automorphism group is given by Aut(X) Aut(X) G. We calculate the group of connected components 0(Aut(X)) by reviewing the classification. A Hopf surface X is a compact complex surface whose universal covering space is W C 2 (0, 0). So, X W G by denoting its covering transformation group by G. A Hopf surface is called primary if its fundamental group G is isomorphic to the group Z of integers, and secondary otherwise. In Theorem 1 of [3] we determined Aut(X) of each secondary Hopf surface X W G so that its holomorphic automorphism group is given by Aut(X) Aut(X) G, where Aut(X) is the normalizer of G in the holomorphic automorphism group of C 2 fixing (0, 0). Moreover, in the following cases (2) and (3) it co incides with the normalizer of G in GL(2, C). We calculate 0(Aut(X)) including primary Hopf surfaces as a continuation by correcting some parts of [3]. Before stating Theorem 2 we review the classification of Hopf surfaces. Except the special case (0) that G is not given by any subgroup of GL(2, C), we may assume that G GL(2, C) and G is an extension of H g G det g 1 by Z. Let a be