Geometric structures on the complement of a projective arrangement

Wim Couwenberg; Gert Heckman; Eduard Looijenga

Publications Mathématiques de l'IHÉS (2005)

  • Volume: 101, page 69-161
  • ISSN: 0073-8301

Abstract

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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.

How to cite

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Couwenberg, Wim, Heckman, Gert, and Looijenga, Eduard. "Geometric structures on the complement of a projective arrangement." Publications Mathématiques de l'IHÉS 101 (2005): 69-161. <http://eudml.org/doc/104212>.

@article{Couwenberg2005,
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.},
author = {Couwenberg, Wim, Heckman, Gert, Looijenga, Eduard},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {hypergeometric; ball-quotient; complex hyperbolic geometry; arrangement; reflection group},
language = {eng},
pages = {69-161},
publisher = {Springer},
title = {Geometric structures on the complement of a projective arrangement},
url = {http://eudml.org/doc/104212},
volume = {101},
year = {2005},
}

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
PB - Springer
VL - 101
SP - 69
EP - 161
AB - 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.
LA - eng
KW - hypergeometric; ball-quotient; complex hyperbolic geometry; arrangement; reflection group
UR - http://eudml.org/doc/104212
ER -

References

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