On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties
by
Michael Kapovich
University of Utah
Coauthors: John J. Millson (University of Maryland)

We prove that for any affine variety S defined over Q there exist an Artin group G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety Hom(G, PO(3))//PO(3). The subset U contains all real points of S. As an application we construct new examples of finitely-presented (Artin) groups which are not fundamental groups of smooth complex algebraic varieties. The proof is based on a scheme-theoretic version of Mnev's universality theorem and on the following generalization of Morgan's test for the fundamental groups of smooth algebraic varieties:

Suppose M is a smooth connected complex algebraic variety with the fundamental group G, L is a reductive algebraic Lie group and f: G --> L is a representation with finite image. Then the germ (Hom(G, L), f) is analytically isomorphic to a quasi-homogeneous cone with generators of weights 1 and 2 and relations of weights 2, 3 and 4. In the case there is a local cross-section through f to Ad(L)-orbits, then the same conclusion is valid for the quotient germ (Hom(G, L)//L, [f]).

http://www.math.utah.edu/~kapovich

Date received: May 29, 1999