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This is a follow up question to a previous question of mine and my thought of answer to it.

Given a (compact) Riemann surface $\Sigma$, a $SL(n,\mathbb{C})$-oper is a rank $n$ holomorphic vector bundle $E\rightarrow \Sigma$, s.th. $\det(E)\cong \mathcal{O}$ is trivial, together with a holomorphic connection $\nabla:E\rightarrow E\otimes K_\Sigma$ ( $K_\Sigma$ being the canonical bundle of $\Sigma$) satisfying the following list of axioms:

1) There is a flitration by holomorphic subbundles $0=F^0\subset F^1\subset \cdots \subset F^n=E$.

2) The connection is Griffiths transverse: $\nabla(F^i)\subset F^{i+1}\otimes K_\Sigma$ .

3) The connection is non-degenerate in the following sense: The induced map $F^i/F^{i-1}\rightarrow F^{i+1}/F^i\otimes K_\Sigma$ is an isomorphism.

By non-degeneracy $E$ is in fact completely specified by $F^1$ and for $SL(n,\mathbb{C})$ it has to hold $F^1\cong K^{\frac{n-1}{2}}_\Sigma$. Having this, an $SL(n, \mathbb{C})$ is equivalent to an $n$-th order differential operator $P:\Omega^{\frac{-n+1}{2}}\rightarrow \Omega^{\frac{n-1}{2}}$, where $\Omega$ is the sheaf of holomorphic sections of $K_\Sigma$.

Q1 Is the converse true? I.e. given a holomorphic line bundle $L\rightarrow \Sigma$ with sheaf of holomorphic sections $\mathcal{L}$ and an n-th order holomorphic differential operator $P:\mathcal{L}\rightarrow \mathcal{L}$, can I construct from this an $SL(n,\mathbb{C})$ oper?

I think to answer to this is yes, but I cannot track a reference for this.

From the discussion I have in my previous question, it seems that to any rank $n$ vector bundle $H$ of degree zero with a holomorphic connection one gets at least locally a $n$-th order holomorphic differential operator defining its flat sections, due to the existence of a local cyclic section.

Q2 As I think the answer to the first question is yes, the local picture cannot glue in general I suppose. So the question is, do the local cyclic sections glue to a global cyclic section if and only if the bundle $H$ has an oper structure?

Improve this question
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    $\begingroup$ I think the answer to your question should be contained, or at least intimated, in these notes of Richard Wentworth: arxiv.org/abs/1402.4203. $\endgroup$
    – Andy Sanders
    May 29, 2020 at 17:39
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    $\begingroup$ See also section 2 of the notes arxiv.org/pdf/math/0501398.pdf of Beilinson-Drinfeld where they also basically answer your question. $\endgroup$
    – Andy Sanders
    May 29, 2020 at 17:41
  • $\begingroup$ @AndySanders, nice, thanks for the references! The beginning of section 2 in the paper by Beilinson-Drinfeld directly answers the first question with the additional assumption that the differential operator has nowhere vanishing principal symbol. So representing the vector bundle locally on a coordinate open as $\mathcal{D}_1(U)/ \mathcal{D}_1P(U)$ the differential operators $P(U)$ only glue to a differential operator with non vanishing principal symbol if the bundle has in fact an oper structure. $\endgroup$
    – mtraube
    May 29, 2020 at 18:39
  • $\begingroup$ Indeed. I'm glad I could say something useful. $\endgroup$
    – Andy Sanders
    May 29, 2020 at 18:41

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