JournalsrlmVol. 17 , No. 2DOI 10.4171/rlm/459

Surface classification and local and global fundamentals groups, I

  • Fabrizio Catanese

    Universität Bayreuth, Germany
Surface classification and local and global fundamentals groups, I cover

Abstract

Given a smooth complex surface SS, and a compact connected global normal crossings divisor D=iDiD = \cup_i D_i, we consider the local fundamental group π1(TD)\pi_1 (T \setminus D) , where TT is a good tubular neighbourhood of DD. One has an exact sequence 1\ra\sK\ra\Ga:=π1(TD)Π:=π1(D)\ra11 \ra \sK \ra \Ga : = \pi_1 (T - D) \rightarrow \Pi : = \pi_1 (D) \ra 1, and the kernel \sK\sK is normally generated by geometric loops \gai\ga_i around the curve DiD_i. Among the main results, which are strong generalizations of a well known theorem of Mumford, is the nontriviality of \gai\ga_i in \Ga=π1(TD)\Ga = \pi_1 (T - D), provided all the curves DiD_i of genus zero have selfintersection Di22D_i^2 \leq -2 (in particular this holds if the canonical divisor KSK_S is nef on DD), and under the technical assumption that the dual graph of DD is a tree.