Geometric Structures on the Complement of a Projective Arrangement

Abstract

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= nite union of hyperplanes) whose Levi- Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements de ned by nite complex re ection groups. We determine a parameter interval for which the metric is locally of Fubini-Study type, at, or complex-hyperbolic. We nd a nite subset of this interval for which we get a complete orbifold or at least a Zariski open subset thereof, and we analyze these cases in some detail (e.g., we determine their orbifold fundamental group). In this set-up, the principal results of Deligne-Mostow on the Lauricella hypergeometric di erential equation and work of Barthel-Hirzebruch-H ofer on arrangements in a projective plane appear as special cases. Along the way we produce in a geometric manner all the pairs of complex re ection groups with isomorphic discriminants, thus providing a uniform approach to work of Orlik-Solomon.