Geometric structures on the complement of a projective arrangement
Publications Mathématiques de l'IHÉS, Volume 101 (2005), pp. 69-161.

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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     title = {Geometric structures on the complement of a projective arrangement},
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Couwenberg, Wim; Heckman, Gert; Looijenga, Eduard. Geometric structures on the complement of a projective arrangement. Publications Mathématiques de l'IHÉS, Volume 101 (2005), pp. 69-161. doi : 10.1007/s10240-005-0032-3. http://www.numdam.org/articles/10.1007/s10240-005-0032-3/

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