Low dimensional projective groups

Details Low dimensional projective groups Low dimensional projective groups

2012-03-20

arxiv.org/abs/1203.4520v3

Mathematics

Geometric Topology

v3: 15 pgs--recast exposition in the general framework of holomorphically convex groups

Scientific paper

We initiate the study of holomorphically convex groups: groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. If $G$ is a holomorphically convex group of cohomological dimension two, we show that $G$ is isomorphic to the fundamental group of a compact Riemann surface. As a consequence, we show that if a linear group $G$ has (rational) cohomological dimension two and is the fundamental group of a smooth complex projective variety, then $G$ is a (virtual) surface group. We also show that if $G$ is a duality group of dimension three, then $G$ cannot be holomorphically convex. As a consequence, if a linear group $G$ is a duality group of dimension three, then $G$ cannot be the fundamental group of a smooth complex projective variety.

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