Construct $b$-symplectic toric manifolds from toric manifolds

TL;DR

This paper introduces a novel method to construct and decompose $b$-symplectic toric manifolds using traditional toric manifolds, enhancing understanding of their structure and relationships.

Contribution

It provides a new construction and decomposition approach for $b$-symplectic toric manifolds based on classical toric manifolds, expanding the toolkit for their analysis.

Findings

Constructs $b$-symplectic toric manifolds from toric manifolds.

Decomposes $b$-symplectic toric manifolds into classical toric manifolds.

Suggests the decomposition is practically useful despite non-canonical structures.

Abstract

In \cite{btoric}, Guillemin et al. proved a Delzant-type theorem which classifies $b$-symplectic toric manifolds. More generally, in \cite{torus} they proved a similar convexity result for general Hamiltonian torus action on $b$-symplectic manifolds. In this paper, we provide a new way to construct $b$-symplectic toric manifolds from usual toric manifolds. Conversely, through this way, we can also decompose a $b$-symplectic toric manifolds to usual toric manifolds. Finally, we will try to prove that this kind of decomposition is useful, although the symplectic structure for our decomposition or construction is not canonical.