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 over M equipped with a holomorphic connection, for all k and n. This provides a description of global holomorphic differential operators from $E\otimes {\mathcal{L}}^{\otimes n}$ to another holomorphic vector bundle F using the natural isomorphism ${\text{Diff}}^k(E\otimes {\mathcal{L}}^{\otimes n},F)=F\otimes {(J^k(E\otimes {\mathcal{L}}^{\otimes n}))}^{⁎}$.

中文翻译：

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黎曼曲面上的射影结构和自然微分算子

我们研究Riemann曲面M上的全纯微分算子。这是通过给M赋予投影结构来完成的。让$\mathcal{大号}$是M上的theta特征。我们明确描述了喷气束${Ĵ}^{ķ}（Ë\otimes {\mathcal{大号}}^{\otimes ñ}）$，其中对于所有k和n，E是在M上配备了全纯连接的全纯矢量束。这提供了对来自$Ë\otimes {\mathcal{大号}}^{\otimes ñ}$使用自然同构到另一个全纯矢量束F${\text{差异}}^{ķ}（Ë\otimes {\mathcal{大号}}^{\otimes ñ}，F）=F\otimes {（{Ĵ}^{ķ}（Ë\otimes {\mathcal{大号}}^{\otimes ñ}））}^{⁎}$。