Surface classification and local and global fundamental 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, i.e., the fundamental group Gamma of T-D, where T is a good tubular neighbourhood of D. One has a surjection of Gamma onto the fundamental group of D, 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 the local fundamental group, provided all the curves $D_i$ of genus zero have selfintersection <= -2. (in particular this holds if the canonical divisor is nef on D), and under the technical assumption that the dual graph of D is a tree.