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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Toric hypersymplectic quotients
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by Andrew Dancer and Andrew Swann PDF
Trans. Amer. Math. Soc. 359 (2007), 1265-1284 Request permission


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:
  • Andrew Swann
  • Affiliation: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
  • MR Author ID: 268267
  • ORCID: 0000-0002-1812-1009
  • Email:
  • Received by editor(s): September 29, 2004
  • Received by editor(s) in revised form: December 21, 2004
  • Published electronically: August 24, 2006
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 1265-1284
  • MSC (2000): Primary 53C25; Secondary 53D20, 53C55, 57S15
  • DOI:
  • MathSciNet review: 2262850