Towards a Classification of Toric and Semitoric Singular Symplectic Manifolds

In this article, we present a classification of toric $b^m$-symplectic manifolds and outline a program for the classification of semitoric singular symplectic manifolds. As a testing ground, we begin with the case of $b$-symplectic manifolds, where a Delzant-type theorem for toric actions has already been established [13]. Building on this foundation, we extend the framework to the broader setting of $b^m$-symplectic manifolds, thereby establishing the first classification result for $b^m$-toric manifolds. This achievement lays the groundwork for a conjectural classification theory of semitoric systems in dimension four on $b^m$-symplectic manifolds. Furthermore, by employing the reduction theory developed in [21], we propose a conjectural classification scheme for higher-dimensional semitoric manifolds, aiming to generalize the Pelayo – Vũ Ngoc classification program [32, 33] to the singular $b^m$-setting.