We develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of K\"ahler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of $\bar{ \partial}$-Laplacians, as well as the convergence result of quantum Hilbert spaces. We also consider the case of almost K\"ahler quantization for compatible almost complex structures, and show the analogous convergence results.
Categories: math.DG math.MG math.SG