Hearing Delzant polytopes from the equivariant spectrum
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Guillemin_Hearing Delzant.pdf
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Author(s)
Dryden, Emily B.
Guillemin, Victor W.
Sena-Dias, Rosa Isabel
Date Issued
October 2011
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
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
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1090/s0002-9947-2011-05412-7
