The geometric quantizations and the measured Gromov-Hausdorff convergences

On a compact symplectic manifold (X,ω) with a prequantum line bundle (L,∇,h), we consider the one-parameter family of ω-compatible complex structures which converges to the real polarization coming from the Lagrangian torus fibration. There are sev-eral 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 ∇,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 S¹-actions, which may depend on the choice of the base point. We observe that the properties of the S¹-actions on the limit spaces actually depend on whether the base point is in the Bohr-Sommerfeld fibers or not.