Surface classification and local and global fundamentals groups, I

Abstract

Given a smooth complex surface $S$, and a compact connected global normal crossings divisor $D={\bigcup }_i{D}_i$, we consider the local fundamental group ${\pi }_1(T\setminus D)$ , where $T$ is a good tubular neighbourhood of $D$. One has an exact sequence $1\to \mathcal{K}\to \Gamma :={\pi }_1(T-D)\to \Pi :={\pi }_1(D)\to 1$, and the kernel $\mathcal{K}$ is normally generated by geometric loops ${\gamma }_i$ around the curve $D_i$. Among the main results, which are strong generalizations of a well known theorem of Mumford, is the nontriviality of ${\gamma }_i$ in $\Gamma ={\pi }_1(T-D)$, provided all the curves $D_i$ of genus zero have selfintersection $D_i^2\le -2$ (in particular this holds if the canonical divisor $K_S$ is nef on $D$), and under the technical assumption that the dual graph of $D$ is a tree.