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 (=a finite union of hyperplanes) whose Levi-Civita connection is of
Dunkl type--interesting examples are obtained from the arrangements defined by
finite complex reflection groups. We determine a parameter interval for which
the metric is locally of Fubini-Study type, flat, or complex-hyperbolic. We
find a finite 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 differential equation and work of Barthel-Hirzebruch-Hoefer on
arrangements in a projective plane appear as special cases. Along the way we
produce in a geometric manner all the pairs of complex reflection groups with
isomorphic discriminants, thus providing a uniform approach to work of
Orlik-Solomon.Comment: 70 pages, v2: We give a better (sharper) formulation of the main
result and added some reference