Convexity for Hamiltonian torus actions on $b$-symplectic manifolds

TL;DR

This paper extends convexity results for moment maps from toric to more general Hamiltonian torus actions on $b$-symplectic manifolds, highlighting the role of modular weights in the behavior of the moment map.

Contribution

It generalizes convexity theorems for moment maps to non-toric Hamiltonian actions on $b$-symplectic manifolds, emphasizing the influence of modular weights.

Findings

When modular weights are zero, the moment map behaves like in classical symplectic geometry.

When modular weights are nonzero, the moment map exhibits behavior similar to the toric $b$-symplectic case.

The modular weights determine the convexity properties of the moment map image.

Abstract

In [GMPS] we proved that the moment map image of a $b$-symplectic toric manifold is a convex $b$-polytope. In this paper we obtain convexity results for the more general case of non-toric hamiltonian torus actions on $b$-symplectic manifolds. The modular weights of the action on the connected components of the exceptional hypersurface play a fundamental role: either they are all zero and the moment map behaves as in classic symplectic one, or they are all nonzero and the moment map behaves as in the toric $b$-symplectic case studied in [GMPS].