Convexity properties of generalized moment maps

TL;DR

This paper investigates the convexity and topological properties of generalized moment maps in the context of Hamiltonian actions on $H$-twisted generalized complex manifolds, extending classical results to a broader geometric setting.

Contribution

It proves convexity and connectedness of generalized moment maps for Hamiltonian torus and Lie group actions on $H$-twisted generalized complex manifolds and orbifolds, extending known convexity theorems.

Findings

Generalized moment map components are Bott-Morse functions.

The image of the generalized moment map is convex.

Fibers of the generalized moment map are connected.

Abstract

In this paper, we consider generalized moment maps for Hamiltonian actions on $H$-twisted generalized complex manifolds introduced by Lin and Tolman \cite{Lin}. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact $H$-twisted generalized complex manifolds and prove that all components of the generalized moment map are Bott-Morse functions. Based on this, we shall show that the generalized moment maps have a convex image and connected fibers. Furthermore, by applying the arguments of Lerman, Meinrenken, Tolman, and Woodward \cite{Ler2} we extend our results to the case of Hamiltonian actions of general compact Lie groups on $H$-twisted generalized complex orbifolds.