On abelian automorphism group of a surface of general type

Abstract

LetS be a minimal surface of general type overℂ,K the canonical divisor ofS. LetG be an abelian automorphism group ofS. IfK 2≧140, then the order ofG is at most 52K 2+32. Examples are also provided with an abelian automorphism group of order 12K 2+96. The automorphism groups for a complex algebraic curve of genusg≧2 have been thoroughly studied by many authors, including many recent ones. In particular, various bounds have been established for the order of such groups: for example, the order of the total automorphism group is ≦84(g−1) [Hu], that of an abelian subgroup is ≦4g+4 [N], while the order of any automorphism is ≦4g+2 ([W], see also [Ha]). It is an intriguing problem to generalise these bounds to higher dimensions. For example, for surfaces of general type, it is well known that the automorphism groups are finite, and the bound of the orders of these groups depends only on the Chern numbers of the surface [A]. In the attempts to such generalisations, the order of abelian subgroups has a special importance. Due to Jordan's theorem on group representations (and its followers), a bound on the order of abelian subgroups induces a bound on that of the whole automorphism group, although bounds thus obtained are generally far from satisfactory. In [H-S], it is shown that for surfaces of general type, the order of such an abelian subgroup is bounded by the square of the Chern numbers times a constant. The purpose of this article is to give a further analysis to the abelian case for surfaces of general type, in proving that the order is bounded linearly by the Chern numbers of the surface, in good analogy with the case of curves. More precisely, our main result is the following.