Solvable Group Representations and Free Divisors Whose Complements Are K(π, 1)‘s

A classical result of Arnold and Brieskorn [Bk], [Bk2] states that the complement of the discriminant of the versal unfolding of a simple hypersurface singularity is a K(π, 1). Deligne [Dg] showed this result could be placed in the general framework by proving that the complement of an arrangement of reflecting hyperplanes for a Coxeter group is again a K(π, 1) (and more generally for simplicial arrangements). A discriminant complement for a simple hypersurface singularity can be obtained as the quotient of the complement of such a hyperplane arrangement by the free action of a finite group, and hence is again is a K(π, 1). What the discriminants and Coxeter hyperplane arrangements have in common is that they are free divisors. This notion was introduced by Saito [Sa], motivated by his discovery that the discriminants for the versal unfoldings of isolated hypersurface singularities are always free divisors. By contrast, Knörrer [Ko] found an isolated complete intersection singularity, for which the complement of the discriminant of the versal unfolding is not a K(π, 1) (although it is again a free divisor by a result of Looijenga [L]). This leads to an intriguing question about when a free divisor has a complement which is a K(π, 1). This remains unsettled for the discriminants of versal unfoldings of isolated hypersurface singularities; this is the classical “K(π, 1)-Problem”. Also, for hyperplane arrangements, there are other families such as arrangements arising from Shephard groups which by Orlik-Solomon [OS] satisfy both properties; however, it remains open whether the conjecture of Saito is true that every free arrangement has complement which is a K(π, 1). A survey of these results on arrangements can be found in the book of Orlik-Terao [OT]. Except for isolated curve singularities in C (and the total space for their equisingular deformations), there are no other known examples of free divisors whose complements are K(π, 1)’s. While neither K(π, 1)-problem has been settled, numerous other classes of free divisors have been discovered so this question continues to arise in new contexts. In this paper, we define a large class of free divisors whose complements are K(π, 1)’s by using the results obtained in [DP1]. These free divisors are “determinantal arrangements”, which are analogous to hyperplane arrangements except that we replace a configuration of hyperplanes by a configuration of determinantal varieties (and the defining equation is a product of determinants rather than a product of linear factors).