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- Title
Non-finiteness properties of fundamental groups of smooth projective varieties.
- Authors
Dimca, Alexandru; Papadima, Ştefan; Suciu, Alexander I.
- Abstract
For each integer n ≧ 2, we construct an irreducible, smooth, complex projective variety M of dimension n, whose fundamental group has infinitely generated homology in degree n + 1 and whose universal cover is a Stein manifold, homotopy equivalent to an infinite bouquet of n-dimensional spheres. This non-finiteness phenomenon is also reflected in the fact that the homotopy group π n( M), viewed as a module over ℤπ1( M), is free of infinite rank. As a result, we give a negative answer to a question of Kollár on the existence of quasi-projective classifying spaces (up to commensurability) for the fundamental groups of smooth projective varieties. To obtain our examples, we develop a complex analog of a method in geometric group theory due to Bestvina and Brady.
- Subjects
FUNDAMENTAL groups (Mathematics); HOMOLOGY theory; STEIN manifolds; MANIFOLDS (Mathematics); HOMOTOPY theory; GROUP theory
- Publication
Journal für die Reine und Angewandte Mathematik, 2009, Vol 2009, Issue 629, p89
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/CRELLE.2009.027