On a compact symplectic manifold ($X, \omega)$ with a prequantum line bundle $(L, \nabla , h)$, we consider the one-parameter family of $\omega$-compatible complex structures which converges to the real polarization coming from the Lagrangian torus fibration. There are several researches which show that the holomorphic sections of the line bundle localize at Bohr–Sommerfeld fibers. In this article we consider the one-parameter family of the Riemannian metrics on the frame bundle of $L$ determined by the complex structures and $\nabla , h$, and we can see that their diameters diverge. If we fix a base point in some fibers of the Lagrangian fibration we can show that they measured Gromov-Hausdorff converge to some pointed metric measure spaces with the isometric $S^1$-actions, which may depend on the choice of the base point. We observe that the properties of the $S^1$-actions on the limit spaces actually depend on whether the base point is in the Bohr–Sommerfeld fibers or not.

中文翻译：

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几何量化和测得的Gromov–Hausdorff收敛

在具有量子线束$（L，\ nabla，h）$的紧实辛流形（$ X，\ omega）$上，我们考虑了兼容$ \ omega $的复杂结构的一参数族来自拉格朗日环面纤维化的极化。有多项研究表明，线束的全同形截面位于Bohr–Sommerfeld纤维处。在本文中，我们考虑了由复杂结构和$ \ nabla，h $所确定的$ L $框架束上的黎曼度量的一参数族，并且我们可以看到它们的直径有所不同。如果我们在拉格朗日纤维的某些纤维中固定一个基点，我们可以证明他们测量的Gromov-Hausdorff会收敛到具有等距$ S ^ 1 $作用的一些尖的度量空间，这可能取决于基点的选择。