  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.
