Hearing Delzant polytopes from the equivariant spectrum

TL;DR

This paper investigates whether the spectrum of the Laplace operator on symplectic toric manifolds can determine their moment polytopes, providing partial positive results under certain genericity and dimensional conditions.

Contribution

It advances the understanding of an equivariant version of Abreu's conjecture, showing spectrum-based determination of moment polytopes in specific cases.

Findings

For 4-dimensional manifolds with generic polygons, the spectrum determines the polygon up to translation.

In higher even dimensions, the equivariant spectrum of the Laplacian on line bundle sections determines the moment polytope.

Results depend on the polygon's genericity and the manifold's topological properties.

Abstract

Let M^{2n} be a symplectic toric manifold with a fixed T^n-action and with a toric K\"ahler metric g. Abreu 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_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.