Automorphism Groups Of Complex Surfaces Of General Type.

It is known that the group of automorphisms of complex surfaces of general type is finite, and in fact bounded by a function of the topological Euler characteristic. The bounding function established by Andreotti is exponential in the Euler characteristic. Since the Hurwitz bound on the automorphism group of a complex curve of general type (i.e. genus at least 2) is linear in the Euler characteristic of the curve, it is natural to ask whether a good bound exists for surfaces of general type as well. In fact, no examples of families of surfaces of general type are known in which the group order grows more than linearly with the Euler characteristic. It is natural to conjecture that the size of the automorphism group is bounded by a polynomial in the Euler characteristic. With the objective of attempting to generalize the proof the Hurwitz bound, the Atiyah-Singer G-theorem is studied. It is shown that the trace formula poses restrictions on the possible local actions. In particular, curves fixed by an automorphism are shown to lie on the vanishing locus of a differential 2-form. Furthermore, the local contributions arising from nontrivial stabilizers are computed: first a reduction is made to stabilizers which are small (i.e. no element acts on the tangent space with an eigenvalue 1), and then a new proof of the continued fraction formula for the contribution is given. The work of Peters on automorphisms acting trivially on the cohomology ring is extended to show that the number of such automorphisms is at most 3 if the algebraic Euler characteristic is larger than 6, at most 4 in general. Since finding a pluricanonical differential form which is invariant under the automorphism group would enable estimation of its size, the plurigenera of the quotient space are computed. Unfortunately, in contrast with the work of Sakai on open surfaces and Fletcher on canonical 3-folds, the existence of an n-canonical form for fixed n cannot be guaranteed. Finally we establish polynomial bounds for a group G acting on a regular surface under the condition that G is solvable or G acts pseudofreely with cyclic stabilizers. The latter estimate is based on studying the 2-genus formula.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128099/2/8801335.pd