Spectral convergence in geometric quantization—the case of non-singular Langrangian fibrations

This paper is a sequel to $\href{https://dx.doi.org/10.4310/JSG.2020.v18.n6.a3}{[11]}$. We develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of Kähler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of $\overline{\partial}$-Laplacians, as well as the convergence result of quantum Hilbert spaces. We also consider the case of almost Kähler quantization for compatible almost complex structures, and show the analogous convergence results.

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K.Hattoriis supported by Grant-in-Aid for Scientific Research (C) Grant Number19K03474. M. Yamashita is supported by Grant-in-Aid for JSPS FellowsGrant Number 19J22404.

Received 9 August 2021

Received revised 3 April 2023

Accepted 13 May 2023

Published 6 June 2024