Semi-Classical Asymptotic Expansions for Toeplitz Quantizations on Complex Manifolds and Orbifolds

TL;DR

This thesis develops asymptotic expansions for Bergman kernels and Toeplitz operators on complex manifolds and orbifolds, advancing the understanding of their spectral and quantization properties.

Contribution

It introduces the local spectral gap condition and derives full asymptotic expansions for Bergman kernels and Toeplitz operators, including deformation quantization.

Findings

Asymptotic expansion for Bergman kernel on orbifolds

Full asymptotic expansion for Toeplitz operators

Establishment of deformation quantization for Toeplitz operators

Abstract

In this thesis, we introduce complex manifolds with local spectral gaps and study their asymptotic behavior using the scaling method. With these asymptotics, we obtain an asymptotic expansion for the Bergman kernel of a Hermitian holomorphic orbifold line bundle satisfying the local spectral gap condition. Furthermore, we establish the full asymptotic expansion of both the Bergman kernel and the Toeplitz operator, using the observations of the scaled Bergman kernel and the stationary phase formula. In addition, we establish the deformation quantization for Toeplitz operators with pseudodifferential operators.