# Cyclic vector of holomorphic vector bundle with flat connection over compact Riemann surface

I originally posted the question on math.stackexhange, but there doesn't seem to be an answer. I apalogize in advance for cross posting.

Let $$E\rightarrow X$$ be a holomorphic vector bundle over a compact Riemann surface with a holomorphic connection $$\nabla:E\rightarrow E\otimes K$$, where $$K$$ is the canonical bundle of $$X$$. Since the holomorphic connection is necessarily flat, its sheaf of local holomorphic sections $$\mathcal{E}$$ defines a (holonomic) $$D$$-module. Every holonomic D-module is locally cyclic, i.e. for any point $$z_0$$ there exists a neighborhood $$U$$ s.th. $$\mathcal{E}(U)$$ has a cyclic generator as a $$D$$-module (see e.g. Proposition 3.1.5. in Björk: Analytic $$D$$-Modules and Applications). Suppose we are given a coordinate $$z$$ on $$U$$ and identify $$D(U)\cong D_1$$, with $$D_1=\mathbb{C}\left\lbrace z \right\rbrace \left\langle \partial_z \right\rangle$$ (differential operators with coefficients in convergent power-series). So locally it holds $$\mathcal{E}(U)\cong D_1/ I$$, where $$I$$ is the ideal of differential operators annihilating the cyclic generator. This ideal is in general generated by two elements $$P,Q$$, with $$P$$ an operator of smallest possible degree in $$I$$ and furthermore $$I/D_1P$$ is of torsion type, i.e. for any $$D\in I$$ it holds $$z^nD\in D_1P$$ for some $$n$$ (Proposition 5.1.4 and Remark 5.1.5 in Björk: Analytic $$D$$-Modules and Applications). This implies for the dual $$D$$-module $$\hom_{D_1}(D_1/I,\mathbb{C}\left\lbrace z\right\rbrace)=\left\lbrace f\in \mathbb{C}\left\lbrace z\right\rbrace \, \middle|\, Pf=Qf=0\right\rbrace=\left\lbrace f\in \mathbb{C}\left\lbrace z\right\rbrace \, \middle|\, Pf=0\right\rbrace$$.

So far so good. Now on $$U$$ the holomorphic connection reads $$\nabla|_U=\partial+A$$ with $$A$$ some matrix of holomorphic functions.The dual bundle naturally comes with a holomorphic connection, too, which in local coordinates takes the form $$\partial-A^T$$. The whole discussion above shows that locally flat sections ($$(\partial-A^T)Y=0$$) are in one to one correspondence with solutions of $$Pf=0$$.

On the other hand there is Deligne's lemma of a cyclic vector. One way to formulate it, is to say that locally on a coordinate neighborhood $$U$$, for a vector bundle with holomorphic connection there exists $$G\in \mathrm{GL}(n,\mathcal{O}(U))$$, s.th. $$\begin{equation} \partial_z G-G A^T=\tilde{A}G \end{equation}$$ with $$\tilde{A}$$ in companion form. Here $$\partial-A^T$$ is the local form of the holomorphic connection. But in general the non unit entries $$a_i$$ in $$\tilde{A}$$ are only meromorphic and $$G$$ might not be invertible as a holomorphic matrix.

It is clear that a system of linear differential equations $$\partial_z Y=A^T Y$$ with $$A^T$$ in companion form corresponds to a single $$n$$-th order scalar differential equation $$Qf=0$$. So from Deligne's cyclic vector lemma I get an $$n$$-th order scalar differential equation, but the corresponding differential operator might not be in $$D_1$$, but in $$\mathbb{C}\left\lbrace z\right\rbrace [z^{-1}]\left\langle \partial_z\right\rangle$$.

Q: Is there any relation between the differential operator I get from the discussion in the first paragraph applied to the dual bundle and the differential equation I get from Deligne's cyclic vector lemma?

I guess they are the same, maybe after imposing further constraints on the open $$U$$. It might very well be that the relation is obvious and just shows my lack of understanding.

I think I found the relation. The paper A Simple Algorithm for Cyclic Vectors by N. Katz seems crucial. The cyclic vector lemma is mostly stated for differential fields (see e.g. section 2 in Galois Theory of Linear Differential Equations), but theorem 1 in the paper by Katz actually doesn't need a field. To be more precise, let $$R$$ be a commutative, local ring with a derivation $$\partial:R\rightarrow R$$, an element $$z\in R$$ s.th. $$\partial(z)=1$$ and $$(V,D,\mathbf{e})$$ a freely, finitely generated $$R$$-module, where $$D$$ is an additive mapping $$D:V\rightarrow V$$ satisfying the usual Leibniz rule $$D(rv)=\partial(r)v+rD(v)$$ and $$\mathbf{e}=(e_0,\dots e_{n-1})$$ is an $$R$$-basis. Then under the assumption that $$R$$ is an $$\mathbb{Z}\left[\frac{1}{(n-1)!}\right]$$-algebra and $$z$$ is in the unique maximal ideal of $$R$$, there is a cyclic vector $$c(\mathbf{e},z)$$, i.e. $$(c,Dc,\dots , D^{n-1}c)$$ is an $$R$$-basis in $$V$$.
Since locally on a trivializing open $$U$$ with coordinate $$z$$, the sheaf of sections $$\mathcal{E}(U)$$ is a free $$\mathbb{C}\left\lbrace z \right\rbrace$$ module and $$\left(\mathbb{C}\left\lbrace z \right\rbrace,\partial_z,z\right)$$ is a local ring in which $$(n-1)!$$ is invertible one can apply the theorem to this situation.
Moreover the path trough the dual bundle seems unnecessary from this. Since $$(c,\nabla_{\frac{\partial}{\partial z}}c,\dots, \nabla^{(n-1)}_{\frac{\partial}{\partial z}}c$$) is a local frame, its corresponding matrix $$C=\left(c\,\nabla_{\frac{\partial}{\partial z}}c\, \cdots\,\nabla^{(n-1)}_{\frac{\partial}{\partial z}}c\right)$$ is invertible as a holomorphic matrix. Applying the gauge transformation $$C^{-1}\left(\partial_z+A\right)C$$ to the local form of the connection gives a connection in companion form and hence an $$n$$-th order scalar differential equation.
On the other hand, from the $$D$$-module point of view, it is clear that there is an equation $$\begin{equation} \nabla^{(n)}_{\frac{\partial}{\partial z}}c=\sum_{i=0}^{n-1}f_i(z)\nabla^{(i)}_{\frac{\partial}{\partial z}}c\end{equation}$$, thus $$P\equiv \partial_z^n-\sum_{i=0}^{n-1}f_i \partial_z^i$$ is in the ideal $$I$$ of $$D_1$$, killing the cyclic generator. In addition I think $$I=D_1P$$, since the leading coefficient in $$P$$ is just $$1$$ and $$\mathcal{E}(U)$$ is locally free $$\mathbb{C}\left\lbrace z\right\rbrace$$-module. Furthermore, $$P$$ is the differential operator one gets from the companion form of the connection.