Room P3.10, Mathematics Building

Rita Pardini, Università di Pisa
The fundamental group of surfaces with small $K^2$

Let $S$ be a smooth minimal complex surface of general type. The numerical invariants of $S$ satisfy the inequalities: $2\chi-6 \leq K^2 \leq 9\chi$. It has now been clear for a long time that the smaller the ratio $K^2/\chi$ is, the simpler the fundamental group of $S$ is. The Severi inequality, which states that the Albanese image of a surface with $K^2<4\chi$ has dimension at most $1$, can be seen as an instance of this general principle. I will show how the Severi inequality and the slope inequality for fibered surfaces can be used to reprove quickly earlier results, due to several authors, on the fundamental group of surfaces with $K^2<3\chi$. Then I will describe recent joint work with Margarida Mendes Lopes, extending these results.