# Surface classification and local and global fundamentals groups, I

### Fabrizio Catanese

Universität Bayreuth, Germany

## 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 $\pi_1 (T \setminus D)$ , where $T$ is a good tubular neighbourhood of $D$. One has an exact sequence $1 \ra \sK \ra \Ga : = \pi_1 (T - D) \rightarrow \Pi : = \pi_1 (D) \ra 1$, 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 $\Ga = \pi_1 (T - D)$, 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.

## 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