A quantization on Riemann surfaces with projective structure

Abstract

Let X be a Riemann surface equipped with a projective structure. Let L be a square-root of the holomorphic cotangent bundle K_x. Consider the symplectic form on the complement of the zero section of L obtained by pulling back the symplectic form on K_x using the map v|→v^⊗2. We show that this symplectic form admits a natural quantization. This quantization also gives a quantization of the complement of the zero section in K_xequipped with the natural symplectic form.