Geometric Quantization Results for Semi-positive Line Bundles on a Riemann Surface

In earlier work (Marinescu and Savale in Math Ann. https://doi.org/10.1007/s00208-023-02750-3 , 2023) the authors proved the Bergman kernel expansion for semi-positive line bundles over a Riemann surface whose curvature vanishes to at most finite order at each point. Here we explore the related results and consequences of the expansion in the semi-positive case including: Tian’s approximation theorem for induced Fubini-Study metrics, leading-order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections, and the asymptotics of holomorphic torsion.