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

Abstract

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.