The geometric quantizations and the measured Gromov-Hausdorff convergences (Q2658223)

The paper under review proposes a new approach to the geometric quantization using the convergence of the Riemannian metrics with respect to the measured Gromov-Hausdorff topology. Fix a prequantum line bundle \(L\) over a compact symplectic manifold \((X, \omega )\). The behaviour of the holomorphic sections of \(L\) are studied here by attaching a family of \(\omega \)-compatible complex structures convergent to the real polarization from the point of view of measured Gromov-Hausdorff convergence. In Section 7 the author proves the pointed \(S^1\)-equivariant measured Gromov-Hausdorff convergence near the Bohr-Sommerfeld fibers of \(L^{\otimes m}\) where \(m\) is a fixed positive integer. A main tool in this study is a condition expressing the existence of a lower bound of the Ricci curvatures. Examples to which these approaches can be applied are discussed in the last section.