Differential operators on a Riemann surface with projective structure (original) (raw)

Let X be a Riemann surface equipped with a projective structure p and L a theta characteristic on X, or in other words, L is a holomorphic line bundle equipped with a holomorphic isomorphism with the holomorphic cotangent bundle Ω X. The complement of the zero section in the total space of the line bundle L has a natural holomorphic symplectic structure, and using p, this symplectic structure has a canonical quantization. Using this quantization, holomorphic differential operators on X are constructed. The main result is the construction of a canonical isomorphism H 0 (X, Diff k X (L ⊗j , L ⊗(i+j+2k)