Automorphisms of multiplicity free Hamiltonian manifolds

TL;DR

This paper proves that multiplicity free Hamiltonian manifolds are uniquely determined by their momentum polytope and isotropy group, by analyzing their automorphism sheaves and cohomology.

Contribution

It computes the automorphism sheaf of such manifolds and shows higher cohomology vanishes, confirming Delzant's conjecture with a key theorem of Losev.

Findings

Automorphism sheaf of multiplicity free Hamiltonian manifolds computed.

Higher cohomology groups of the sheaf are shown to vanish.

Confirms Delzant's conjecture on the uniqueness of these manifolds.

Abstract

We compute the sheaf of automorphisms of a multiplicity free Hamiltonian manifold over its momentum polytope and show that its higher cohomology groups vanish. Together with a theorem of Losev, arXiv:math/0612561, this implies a conjecture of Delzant: a compact multiplicity free Hamiltonian manifold is uniquely determined by its momentum polytope and its principal isotropy group.