Abstract:
Consider a complex projective space with its Fubini-Study metric.
We study one parameter deformations of this metric on the complement
of an arrangement (= finite union of hyperplanes) whose Levi-Civita
connection is of Dunkl type. We concentrate on the parameters for which the
metric is locally of Fubini-Study type, flat, or complex-hyperbolic.
In these cases we study the associated developing map and then focus
on the cases for which we get orbifolds or where there is a Baily-Borel
compactification (this happens for discrete parameters only). Interesting
examples are obtained from the arrangements defined by finite
complex reflection groups. The principal result 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. This is also a setting which produces in a geometric manner
all the pairs of complex reflection groups with isomorphic discriminants,
and thus provides a uniform approach to work of Orlik-Solomon.