Publications mathématiques de l'IHÉS

, Volume 101, Issue 1, pp 69–161 | Cite as

Geometric Structures on the Complement of a Projective Arrangement

Article

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 (= 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-Höfer 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.

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Copyright information

© Institut des Hautes Études Scientifiques and Springer-Verlag 2005

Authors and Affiliations

  1. 1.Océ Technologies BVVenloThe Netherlands
  2. 2.Mathematisch InstituutRadboud UniversiteitNijmegenThe Netherlands
  3. 3.Faculteit Wiskunde en InformaticaUniversiteit UtrechtUtrechtThe Netherlands

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