Geometric structures on the complement of a projective arrangement
摘要
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 (=a 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-Hoefer 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.
引用
@article{arxiv.math/0311404,
title = {Geometric structures on the complement of a projective arrangement},
author = {Wim Couwenberg and Gert Heckman and Eduard Looijenga},
journal= {arXiv preprint arXiv:math/0311404},
year = {2007}
}
备注
70 pages, v2: We give a better (sharper) formulation of the main result and added some references