JournalsrlmVol. 17, No. 2pp. 135–153

Surface classification and local and global fundamentals groups, I

  • Fabrizio Catanese

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

Cite this article

Fabrizio Catanese, Surface classification and local and global fundamentals groups, I. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), no. 2, pp. 135–153

DOI 10.4171/RLM/459