Surface classification and local and global fundamentals groups, I

Abstract

Given a smooth complex surface $S$, and a compact connected global normal crossings divisor $D = \cup_i D_i$, 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 $1 \ra \sK \ra \Ga : = \pi_1 (T - D) \rightarrow \Pi : = \pi_1 (D) \ra 1$, and the kernel $\sK$ is normally generated by geometric loops $\ga_i$ around the curve $D_i$. Among the main results, which are strong generalizations of a well known theorem of Mumford, is the nontriviality of $\ga_i$ in $\Ga = \pi_1 (T - D)$, provided all the curves $D_i$ of genus zero have selfintersection $D_i^2 \leq -2$ (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.