Polynomial bounds for automorphism groups offoliations

Abstract

This thesis concerns the problem of bounding polynomially orders of automorphism groups foliated surfaces (X;F) of general type. After a brief recall of useful properties of foliated surfaces in the first chapter, the second one deals with the local behavior of such groups around a fixed point. Results are stated for regular points and reduced singularities. However, the approach is useful to describe the non-reduced case, as remarked at the end of the chapter. In the third chapter, foliations on the projective plane P2 are studied. It is proved that the automorphism group of a degree d foliation can be bounded quadratically on d. This bound is sharp, it is attained by the automorphism groups of the Jouanolou's examples. The fourth chapter concerns the case when X is geometrically ruled. Results toward the classication of foliations on such surfaces are stated and, under mild hypotheses, quadratic bounds are given. The last chapter is about foliations on surfaces that are not birationally ruled. Regular foliations of general type are analyzed. It is proved that they live on minimal surfaces of general type, hence linear bounds are trivially obtained. Next, singular foliations with ample canonical bundle arestudied and cubic bounds are given for their automorphism groups undermild restrictions. Finally, it is shown that the technique developed so farprovide bounds for the automorphism groups of general type foliations with a holomorphic first integral.