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Let me state it using the language of $D$-modules. Let $X$ be a smooth projective curve and $\mathcal{L}$ a line bundle on it. Assume that $\mathcal{L}$ has a left action of $\mathcal{D}_X$. Then show that $\mathrm{deg}\, \mathcal{L} = 0$.

This is true for $\mathcal{L}=\mathcal{O}_X$ clearly. And this question is purely algebraic. Though it has a proof based in analytic method over $\mathbb{C}$, I would like to ask for a proof in algebraic method, maybe available in positive characteristic.

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    $\begingroup$ If $F:C\to C$ is the absolute Frobenius morphism and $C$ is a proper smooth curve over the finite field ${\bf F}_p$, then for any line bundle $L$ on $C$, there is a connection on $F^*(L)=L^{\otimes p}$ but in general one does not have ${\rm deg}(F^*(L))=0$. I think one needs some form of the Riemann-Hilbert correspondence to prove the statement you consider in char. $0$, so analytic methods might be required. $\endgroup$ Oct 26, 2021 at 14:21
  • $\begingroup$ see Biswas article dx.doi.org/10.1007/s10977-005-4708-8 which deals with vector bundles, also in positive characteristic $\endgroup$
    – Niels
    Oct 26, 2021 at 14:25
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    $\begingroup$ $c_1(L)\in H^1(X, \Omega^1)$ is $\pm$ the class of the $\Omega^1$-torsor of connections on $L$. So, this class is zero if, and only if, $L$ admits a connection. For a smooth projective curve $H^1(X, \Omega^1) = k$ and the image of $c_1(L)$ is the image of $\deg(L)$ under the map $\mathbb{Z}\rightarrow k$. In positive characteristic this means the degree has to be divisible by $p$, explaining Damian Rössler's example. $\endgroup$ Oct 26, 2021 at 14:27
  • $\begingroup$ see also THE ATIYAH-WEIL CRITERION FOR HOLOMORPHIC CONNECTIONS Indranil Biswas and N. Raghavendra insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20014c96_3.pdf $\endgroup$
    – Niels
    Oct 26, 2021 at 14:31
  • $\begingroup$ @Pavel Safronov. This is interesting. I did not know that one could describe the first Chern class in Hodge cohomology in this way. Where is this described? $\endgroup$ Nov 1, 2021 at 10:04

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