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

 

Hearing Delzant polytopes from the equivariant spectrum


Authors: Emily B. Dryden, Victor Guillemin and Rosa Sena-Dias
Journal: Trans. Amer. Math. Soc. 364 (2012), 887-910
MSC (2010): Primary 58J50, 53D20
Published electronically: October 4, 2011
MathSciNet review: 2846357
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05412-7
PII: S 0002-9947(2011)05412-7
Keywords: Laplacian, symplectic manifold, toric, Delzant polytope, equivariant spectrum
Received by editor(s): August 24, 2009
Received by editor(s) in revised form: June 18, 2010
Published electronically: October 4, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.