Convexity and Thimm's Trick
Abstract
In this paper we prove a convexity and fibre-connectedness theorem for proper maps constructed by Thimm's trick on a connected Hamiltonian -space that generate a Hamiltonian torus action on an open dense submanifold. Since these maps only generate a Hamiltonian torus action on an open dense submanifold of , convexity and fibre-connectedness do not follow immediately from Atiyah-Guillemin-Sternberg's convexity theorem, even if is compact. The core contribution of this paper is to provide a simple argument circumventing this difficulty. In the case where the map is constructed from a chain of subalgebras we prove that the image is given by a list of inequalities that can be computed explicitly. This generalizes the famous example of Gelfand-Zeitlin systems on coadjoint orbits introduced by Guillemin and Sternberg. Moreover, we prove that if such a map generates a completely integrable torus action on an open dense submanifold of , then all its fibres are smooth embedded submanifolds.
Cite
@article{arxiv.1509.07356,
title = {Convexity and Thimm's Trick},
author = {Jeremy Lane},
journal= {arXiv preprint arXiv:1509.07356},
year = {2021}
}
Comments
This final version of the paper is to appear. The introduction has changed to better explain the relevance of this work. A brief section comparing the contents of this paper to symplectic contraction recently introduced by HMM has been reinstated. The details of several examples are expanded. A gap in an earlier proof of Proposition 4 has been corrected, thanks to the keen eye of the referees