Differential operators on a Riemann surface with projective structure

Abstract

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 ΩX. 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 H⁰(X,Diff^kX(L^⊗j,L^⊗(i+j+2k)))≅⊕_l=0^kH⁰(X,L^⊗i⊗ Ω_x^⊗l), i,j∈Z, n≤ 0, provided i∉[-2(k-1),0].