2025/07/20 by Y.H. Tsai, Tsai, Yi-Hsin
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2508.00008
openalex publication_date 2025/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.