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

@article{PMIHES_2005__101__69_0,
     author = {Couwenberg, Wim and Heckman, Gert and Looijenga, Eduard},
     title = {Geometric structures on the complement of a projective arrangement},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer},
     volume = {101},
     year = {2005},
     pages = {69-161},
     doi = {10.1007/s10240-005-0032-3},
     zbl = {1083.14039},
     mrnumber = {2217047},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2005__101__69_0}
}
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/item/PMIHES_2005__101__69_0/

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