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 Ngọc classification program [32, 33] to the singular $$b^{m}$$ -setting.