Surfaces of general type with extremal cotangent dimension

RESUMO

We study the geography of surfaces of general type with extremal cotangent dimension, in other words surfaces which have either no global holomorphic symmetric differentials at all or the maximal asymptotic growth of their number, i.e. big cotangent bundle. We are mainly interested in surfaces with low slope K^2 /$\chi$, which, as far as maximal cotangent dimension is concerned, were out of reach of previous methods. We prove vanishing theorems for symmetric logarithmic differentials on minimal rational surfaces and for differentials on their double covers and extend a bigness criterion of Sakai to fibrations of general type in the sense of Campana. As a consequence, we prove, on the one hand that generic Horikawa surfaces have no nontrivial symmetric differentials and on the other hand that there exist Horikawa surfaces with big cotangent bundle.