Spectral convergence in geometric quantization --- the case of toric symplectic manifolds

Abstract

In this paper, we show the spectral convergence result of $\overline{\partial }$-Laplacians when $(X,\omega )$ is a compact toric symplectic manifold equipped with the natural prequantum line bundle $L$. We consider a family $\{J_s{\}}_s$ of $\omega$-compatible complex structures tending to the large complex structure limit, and obtain the spectral convergence of $\overline{\partial }$-Laplacians acting on $L^k$.