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) \rightarrow \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 \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.