Geometric Structures on the Complement of a Projective Arrangement
By W. Couwenberg, G. Heckman, E. Looijenga. 2003
67 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/5c14924014d802a360b33d80d71afb4d28cbed3e
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
- David Bessis
- Igor Dolgachev
- Nathan Reading
- Drew Armstrong
- Serge Cantat
- Michelle L. Wachs
- Jon McCammond
- Tom D. Brady
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Reflection groups in algebraic geometry (2006, methodology): “We refer to [25] for a construction of complex reflection subgroups of finite volume which are not commensurable to the groups from the Deligne-Mostow list.”
- Finite complex reection arrangements are K(; 1) (2015, background): “All references and notations are from [22].”
- Finite complex reflection arrangements are $K(\pi,1)$ (2006, unknown): “The work of Couwenberg-Heckman-Looijenga [25] could possibly be adapted to provide a direct argument.”