A Quantization on Riemann Surfaces with Projective Structure

Abstract

Let X be a Riemann surface equipped with a projective structure. Let $\mathcal{L}$ be a square-root of the holomorphic cotangent bundle K _X . Consider the symplectic form on the complement of the zero section of $\mathcal{L}$ obtained by pulling back the symplectic form on K _X using the map ν ↦ ν^⊗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 _X equipped with the natural symplectic form.