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},
pages = {69--161},
year = {2005},
publisher = {Springer},
volume = {101},
doi = {10.1007/s10240-005-0032-3},
mrnumber = {2217047},
zbl = {1083.14039},
language = {en},
url = {https://www.numdam.org/articles/10.1007/s10240-005-0032-3/}
}
TY - JOUR AU - Couwenberg, Wim AU - Heckman, Gert AU - Looijenga, Eduard TI - Geometric structures on the complement of a projective arrangement JO - Publications Mathématiques de l'IHÉS PY - 2005 SP - 69 EP - 161 VL - 101 PB - Springer UR - https://www.numdam.org/articles/10.1007/s10240-005-0032-3/ DO - 10.1007/s10240-005-0032-3 LA - en ID - PMIHES_2005__101__69_0 ER -
%0 Journal Article %A Couwenberg, Wim %A Heckman, Gert %A Looijenga, Eduard %T Geometric structures on the complement of a projective arrangement %J Publications Mathématiques de l'IHÉS %D 2005 %P 69-161 %V 101 %I Springer %U https://www.numdam.org/articles/10.1007/s10240-005-0032-3/ %R 10.1007/s10240-005-0032-3 %G en %F 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, Tome 101 (2005), pp. 69-161. doi: 10.1007/s10240-005-0032-3
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