The so-called quantization problem in geometric quantization is asking whether the space of wave functions is independent of the choice of polarization. In this paper, we apply SYZ transforms to solve the quantization problem in two cases: (1) semi-flat Lagrangian torus fibrations over complete compact integral affine manifolds, and (2) projective toric manifolds. More precisely, we prove that the space of wave functions associated to the real polarization is canonically isomorphic to that associated to a complex polarization via SYZ transforms in both cases.

中文翻译：

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通过 SYZ 变换进行几何量化

几何量化中所谓的量化问题就是问波函数的空间是否与极化的选择无关。在本文中，我们应用 SYZ 变换来解决两种情况下的量化问题：（1）完全紧凑积分仿射流形上的半平坦拉格朗日环面纤维化，以及（2）投影复曲面流形。更准确地说，我们证明了与实极化相关的波函数空间在两种情况下都与通过 SYZ 变换与复极化相关的波函数空间规范同构。