The geometric quantizations and the measured Gromov-Hausdorff convergences

TL;DR

This paper studies the geometric quantization of symplectic manifolds, showing how certain metric spaces converge in the measured Gromov-Hausdorff sense as complex structures vary, revealing dependence on Bohr-Sommerfeld fibers.

Contribution

It introduces a new perspective on the convergence of metric measure spaces associated with geometric quantization, highlighting the role of Bohr-Sommerfeld fibers in the limit behavior.

Findings

Measured Gromov-Hausdorff convergence to metric spaces with $S^1$-actions

Limit space properties depend on whether base points are in Bohr-Sommerfeld fibers

Diameters of certain metrics diverge as complex structures vary

Abstract

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.