Projective structures on Riemann surface and natural differential operators

Abstract

We investigate the holomorphic differential operators on a Riemann surface $M$. This is done by endowing $M$ with a projective structure. Let $\mathcal{L}$ be a theta characteristic on $M$. We explicitly describe the jet bundle $J^k(E\otimes {\mathcal{L}}^{\otimes n})$, where $E$ is a holomorphic vector bundle on $M$ equipped with a holomorphic connection, for all $k$ and $n$. This provides a description of holomorphic differential operators from $E\otimes {\mathcal{L}}^{\otimes n}$ to another holomorphic vector bundle $F$ using the natural isomorphism ${\text{Diff}}^k(E\otimes {\mathcal{L}}^{\otimes n},F)=F\otimes (J^k(E\otimes {\mathcal{L}}^{\otimes n}){)}^{\ast }$.

Final version; to appear in "Differential Geometry and its Applications"