Best Bounds On Abelian Automorphism Groups Of Surfaces Of General Type

The main purpose of this paper is to give the best upper bound of an abelian automorphism group of a complex algebraic surface of general type.Let S be a minimal smooth complex surface of general type, and G be an abelian automorphism group of S, i.e., an abelian subgroup of the automorphism group Aut(S) of S. We prove that the order|G|of G is bounded from above by12.5K2S+100provided the geometric genus of S is greater that6, where K2S is the first Chern number of S. Furthermore, if G is cyclic, then|G|≤12.5K2S+90. The upper bounds can be reached for an infinite number of examples of surfaces where the geometric genus is arbitrarily large. We construct also a surface S with geometric genus3, and S admits an abelian automorphism group G of order|G|=12.5K2S+103.The first step of our proofs is to analyze the induced actions of G on the spaces of global sections of the pluri-canonical line bundles of the surface. Then the problem is reduced to the case where the surface admits a fibration f:Sâ†'B, and G preserves the fibers of the fibration, i.e., every automorphism in G maps a fiber to a fiber, G C Aut(f), where Aut(f) C Aut(S) is the automorphism subgroup preserving the fibration f:Sâ†'B.The second step is to bound from above the order of the abelian subgroups G of Aut(f). The difficult case is when B(?)P1. Let f:Sâ†'P1be a relatively minimal fibration of genus g≥2, and G C Aut(f) an abelian automorphism subgroup. If f admits at least3singular fibers, then we prove that|G|≤max{2(g+1)/g-1K2f,2g+1/gef}≤(2g+1)2/g(g-1)K2f: where K2f and ef are respectively the relative Chern numbers of the fibration f. The upper bounds here are all optimal.One of our new techniques in the proofs is that we consider some middle quotient maps Sâ†'S/K for various subgroups K of G, instead of the total quotient map Sâ†'S/G only as usual. The advantage of this method is that we can control simultaneously the ramification divisors of the quotient maps and the order of the groups by choosing various subgroups K. Our second new technique is to use the base change method. We prove that the abelian automorphism group G of the fibration can be lifted under some proper base changes. On the other hand, we know the relationships between the old and new relative Chern numbers under a base change by using the formulas for the modular invariants of a fibration. Then we let the degrees of the base changes tend to infinite, the limit case of the inequalities give us the optimal inequalities we desired. We use also the base change technique to construct some examples of fibrations with some abelian automorphism groups whose orders reach the upper bounds, and the geometric genera can be arbitrarily large.As an application of our techniques, we obtain some optimal upper bounds of an abelian automorphism group G of a d-gonal algebraic curve of genus g≥2. We prove that|G|≤2g/d-1g+2d except the curve is a plane Fermat curve of degree d+1. Furthermore, if G is cyclic, then|G|≤2d/g-1g+d. For every d≥2and g≥2, the bounds are sharp. Because d≤[g+3/2], we see that the inequalities imply the classic upper bounds|G|≤4g+4, or|G|≤4g+2when G is cyclic.