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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Toric hypersymplectic quotients


Authors: Andrew Dancer and Andrew Swann
Journal: Trans. Amer. Math. Soc. 359 (2007), 1265-1284
MSC (2000): Primary 53C25; Secondary 53D20, 53C55, 57S15
Published electronically: August 24, 2006
MathSciNet review: 2262850
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Abstract: We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space $ \mathbb{R}^{4d}$ by the action of a compact Abelian group. These $ 4n$-dimensional quotients carry a multi-Hamilitonian action of an $ n$-torus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in $ \mathbb{R}^{3n}$. We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial structures on $ \mathbb{R}^{4n}$ and metrics on complements of hypersurfaces in compact manifolds.


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Additional Information

Andrew Dancer
Affiliation: Jesus College, Oxford, OX1 3DW, United Kingdom
Email: dancer@maths.ox.ac.uk

Andrew Swann
Affiliation: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
Email: swann@imada.sdu.dk

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03925-0
PII: S 0002-9947(06)03925-0
Keywords: Hypersymplectic structure, neutral hyperk\"ahler manifold, toric variety, moment map
Received by editor(s): September 29, 2004
Received by editor(s) in revised form: December 21, 2004
Published electronically: August 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.