Vv. Benyashkrivets et Vi. Chernousov, REPRESENTATION VARIETIES OF THE FUNDAMENTAL-GROUPS OF COMPACT NONORIENTABLE SURFACES, Sbornik. Mathematics, 188(7-8), 1997, pp. 997-1039
Let Gamma(g) be the fundamental group of a compact non-orientable surf
ace of genus g and let K be an algebraically closed field of character
istic O. The structure of the representation varieties R(Gamma(g), GL(
n)(K)), R(Gamma(g),SLn(K)) of Gamma(g) into GL,(K) and SLn(K) and of t
he character varieties X(Gamma(g),GL(n)(K)) is described; namely, the
number of their irreducible components and their dimensions are determ
ined and their birational properties are investigated. It is proved, i
n particular, that all the irreducible components of R(Gamma(g),GL(n)(
K)) are Q-rational varieties.