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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Hearing Delzant polytopes from the equivariant spectrum
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by Emily B. Dryden, Victor Guillemin and Rosa Sena-Dias PDF
Trans. Amer. Math. Soc. 364 (2012), 887-910 Request permission

Abstract:

Let $M^{2n}$ be a symplectic toric manifold with a fixed $\mathbb {T}^n$-action and with a toric Kähler metric $g$. Abreu (2003) asked whether the spectrum of the Laplace operator $\Delta _g$ on $\mathcal {C}^\infty (M)$ determines the moment polytope of $M$, and hence by Delzant’s theorem determines $M$ up to symplectomorphism. We report on some progress made on an equivariant version of this conjecture. If the moment polygon of $M^4$ is generic and does not have too many pairs of parallel sides, the so-called equivariant spectrum of $M$ and the spectrum of its associated real manifold $M_{\mathbb {R}}$ determine its polygon, up to translation and a small number of choices. For $M$ of arbitrary even dimension and with integer cohomology class, the equivariant spectrum of the Laplacian acting on sections of a naturally associated line bundle determines the moment polytope of $M$.
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Additional Information
  • Emily B. Dryden
  • Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
  • Email: ed012@bucknell.edu
  • Victor Guillemin
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • Email: vwg@math.mit.edu
  • Rosa Sena-Dias
  • Affiliation: Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
  • Email: senadias@math.ist.utl.pt
  • Received by editor(s): August 24, 2009
  • Received by editor(s) in revised form: June 18, 2010
  • Published electronically: October 4, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 887-910
  • MSC (2010): Primary 58J50, 53D20
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05412-7
  • MathSciNet review: 2846357