In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [13] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on \(T^*(SO(3))\) to a sphere. Our geometric quantization formulation coincides with the models given in [11] and [21] away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [4]. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general.

在本文中，我们给出了几何量化的通用模型，该模型与具有非退化奇点的可积系统给出的真实极化相关。这个通用模型比 [13] 中的余切模型更进一步，既考虑了奇异轨道，又向余切模型添加了前量子线丛的模型。这些奇点在由莫尔斯型函数给出的意义上是通用的，包括椭圆、双曲线和焦点聚焦奇点。承认这种奇点的系统的例子是复曲面、半复曲面和几乎复曲面流形，以及物理系统，例如谐振子的耦合、球摆或刚体在\(T^*(SO (3))\)到一个球体。我们的几何量化公式与 [11] 和 [21] 中给出的模型相吻合，远离奇点，并修正了以前的双曲奇点和焦点聚焦奇点模型，抵消了以前方法获得的无限维贡献。此处提供的几何量化模型与机械系统的经典物理方法相匹配，例如 [4] 中介绍的球摆。我们的余切模型遵循局部到全局原理，即使系统的全局辛分类一般未知，也可以粘合以确定全局系统的几何量化。