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Hearing Delzant polytopes from the equivariant spectrum

Author(s)
Dryden, Emily B.; Guillemin, Victor W.; Sena-Dias, Rosa Isabel
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Abstract
Let M[superscript 2n] be a symplectic toric manifold with a fixed T[superscript n]-action and with a toric Kähler metric g. Abreu (2003) asked whether the spectrum of the Laplace operator Δ[subscript g] on C∞ (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[superscript 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[subscript 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.
Description
Author's final manuscript June 18, 2012
Date issued
2011-10
URI
http://hdl.handle.net/1721.1/80861
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Transactions of the American Mathematical Society
Publisher
American Mathematical Society
Citation
Dryden, Emily B., Victor Guillemin, and Rosa Sena-Dias. “Hearing Delzant polytopes from the equivariant spectrum.” Transactions of the American Mathematical Society 364, no. 2 (February 1, 2012): 887-910.
Version: Author's final manuscript
ISSN
0002-9947
1088-6850

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