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Zbl 1170.14017
Dimca, Alexandru; Papadima, Ştefan; Suciu, Alexander I.
Non-finiteness properties of fundamental groups of smooth projective varieties. (English)
[J] J. Reine Angew. Math. 629, 89-105 (2009). ISSN 0075-4102; ISSN 1435-5345

A {\it projective group} is the fundamental group of a smooth complex projective variety. \par This paper investigates the finiteness properties of projective groups, mainly motivated by a question posed by {\it J. Kollár} [Shafarevich maps and automorphic forms. Princeton, NJ: Princeton University Press (1995; Zbl 0871.14015)]: {\it Is a projective group $G$ commensurable (up to finite kernels) with another group $G'$ admitting a $K(G',1)$ which is a quasi-projective variety?} \par The main result of this paper is that the answer of Kollár's question is negative. In fact the authors prove the existence, for each $n \geq 2$, of a smooth irreducible complex projective variety of dimension $n$ and fundamental group not commensurable (up to finite kernels) to any group having a classifying space of finite type. These varieties have the homotopy type of an infinite bouquet of $n-$dimensional spheres, and their universal cover is a Stein manifold (which is very interesting because of the Shafarevich conjecture). \par To obtain the result the authors develop a complex analog of the Bestvina-Brady method in geometric group theory [{\it M. Bestvina} and {\it N. Brady}, Invent. Math. 129, No. 3, 445--470 (1997; Zbl 0888.20021)] well adapted to construct projective groups with controlled finiteness properties.
[Roberto Pignatelli (Trento)]

MSC 2000:: *14F35 Homotopy theory (algebraic geometry)

Keywords: Projective groups; Shafarevich conjecture

Citations: Zbl 0871.14015; Zbl 0888.20021

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