Projective structures on Riemann surface and natural differential operators

TL;DR

This paper studies holomorphic differential operators on Riemann surfaces with projective structures, explicitly describing jet bundles and their relation to differential operators using natural isomorphisms.

Contribution

It provides explicit descriptions of jet bundles for vector bundles with connections on Riemann surfaces endowed with a projective structure, linking them to differential operators.

Findings

Explicit formulas for jet bundles $J^k(E\otimes \mathcal{L}^{\otimes n})$

Characterization of differential operators via jet bundle isomorphisms

Application to holomorphic differential operators on Riemann surfaces

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 $\text{Diff}^k(E\otimes {\mathcal L}^{\otimes n}, F)= F\otimes (J^k(E\otimes {\mathcal L}^{\otimes n}))^*$.