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Geometric Structures on the Complement of a Projective Arrangement

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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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Authors and Affiliations

  1. Océ Technologies BV, P.O. Box 101, NL-5900 MA, Venlo, The Netherlands

    Wim Couwenberg

  2. Mathematisch Instituut, Radboud Universiteit, P.O. Box 9010, NL-6500 GL, Nijmegen, The Netherlands

    Gert Heckman

  3. Faculteit Wiskunde en Informatica, Universiteit Utrecht, P.O. Box 80.010, NL-3508 TA, Utrecht, The Netherlands

    Eduard Looijenga

Authors
  1. Wim Couwenberg
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  2. Gert Heckman
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  3. Eduard Looijenga
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Correspondence to Wim Couwenberg, Gert Heckman or Eduard Looijenga.

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In memory of Peter Slodowy (1948–2002)

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Couwenberg, W., Heckman, G. & Looijenga, E. Geometric Structures on the Complement of a Projective Arrangement. Publ.math.IHES 101, 69–161 (2005). https://doi.org/10.1007/s10240-005-0032-3

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