A General Local-to-Global Principle for Convexity of Momentum Maps

TL;DR

This paper generalizes the local-to-global principle for momentum map convexity, allowing for broader target spaces and removing some traditional assumptions, thus extending the scope of convexity theorems.

Contribution

It introduces a new factorization of the momentum map and a unified approach to convexity, applicable to non-closed maps and spaces without metric-based convexity.

Findings

Convexity of momentum map images is established without local fiber connectedness.

The approach applies to a wide class of spaces, not necessarily metric-based.

A new method of geodesic construction via straightening enhances previous results.

Abstract

We extend the Local-to-Global-Principle used in the proof of convexity theorems for momentum maps to not necessarily closed maps whose target space carries a convexity structure which need not be based on a metric. Using a new factorization of the momentum map, convexity of its image is proved without local fiber connectedness, and for almost arbitrary spaces of definition. Geodesics are obtained by straightening rather than shortening of arcs, which allows a unified treatment and extension of previous convexity results.