Hearing Delzant polytopes from the equivariant spectrum
Author(s)Dryden, Emily B.; Guillemin, Victor W.; Sena-Dias, Rosa Isabel
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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.
Author's final manuscript June 18, 2012
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Transactions of the American Mathematical Society
American Mathematical Society
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.
Author's final manuscript