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.

20 pages, to appear in the Rendiconti Accademia Lincei, new series, volume dedicated to the 90-th birthday of Guido Zappa. The paper is quite appropriately dedicated to Zappa, who helped the author with an important suggestion to consider certain equivalence classes of words. The revision corrects some typo and contains one more bibliographical reference