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ca.classical evaluation and odes – Cyclic vector of holomorphic vector bundle with flat connection over compact Riemann floor retort

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ca.classical evaluation and odes – Cyclic vector of holomorphic vector bundle with flat connection over compact Riemann floor

I initially posted the query on math.stackexhange, however there does not emerge to breathe an retort. I apalogize in close by for nasty posting.

Let $Erightarrow X$ breathe a holomorphic vector bundle over a compact Riemann floor with a holomorphic connection $nabla:Erightarrow Eotimes Okay$, the place $Okay$ is the canonical bundle of $X$. Since the holomorphic connection is essentially flat, its sheaf of native holomorphic sections $mathcal{E}$ defines a (holonomic) $D$-module. Every holonomic D-module is regionally cyclic, i.e. for any level $z_0$ there exists a neighborhood $U$ s.th. $mathcal{E}(U)$ has a cyclic generator as a $D$-module (graze e.g. Proposition 3.1.5. in Björk: Analytic $D$-Modules and Applications). Suppose we’re given a coordinate $z$ on $U$ and establish $D(U)cong D_1$, with $D_1=mathbb{C}leftlbrace z properrbrace leftlangle partial_z properrangle$ (differential operators with coefficients in convergent power-succession). So regionally it holds $mathcal{E}(U)cong D_1/ I$, the place $I$ is the mannequin of differential operators annihilating the cyclic generator. This mannequin is in widespread generated by two components $P,Q$, with $P$ an operator of smallest workable diploma in $I$ and moreover $I/D_1P$ is of torsion kindly, i.e. for any $Din I$ it holds $z^nDin D_1P$ for some $n$ (Proposition 5.1.4 and Remark 5.1.5 in Björk: Analytic $D$-Modules and Applications). This implies for the twin $D$-module $hom_{D_1}(D_1/I,mathbb{C}leftlbrace zproperrbrace)=leftlbrace fin mathbb{C}leftlbrace zproperrbrace , center|, Pf=Qf=0properrbrace=leftlbrace fin mathbb{C}leftlbrace zproperrbrace , center|, Pf=0properrbrace$.

So far so glorious. Now on $U$ the holomorphic connection reads $nabla|_U=partial+A$ with $A$ some matrix of holomorphic capabilities.The twin bundle naturally comes with a holomorphic connection, too, which in native coordinates takes the design $partial-A^T$. The all dialogue above reveals that regionally flat sections ($(partial-A^T)Y=0$) are in a single to at least one correspondence with options of $Pf=0$.

On the opposite hand there may be Deligne’s lemma of a cyclic vector. One path to formulate it, is to say that regionally on a coordinate neighborhood $U$, for a vector bundle with holomorphic connection there exists $Gin mathrm{GL}(n,mathcal{O}(U))$, s.th.
commence{equation} partial_z G-G A^T=tilde{A}G stop{equation} with $tilde{A}$ in companion design. Here $partial-A^T$ is the native design of the holomorphic connection. But in widespread the non unit entries $a_i$ in $tilde{A}$ are solely meromorphic and $G$ vitality not breathe invertible as a holomorphic matrix.

It is limpid {that a} system of linear differential equations $partial_z Y=A^T Y$ with $A^T$ in companion design corresponds to a sole $n$-th bid scalar differential equation $Qf=0$. So from Deligne’s cyclic vector lemma I purchase an $n$-th bid scalar differential equation, however the corresponding differential operator vitality not breathe in $D_1$, however in $mathbb{C}leftlbrace zproperrbrace [z^{-1}]leftlangle partial_zproperrangle$.

Q: Is there any relation between the differential operator I purchase from the dialogue within the first paragraph utilized to the twin bundle and the differential equation I purchase from Deligne’s cyclic vector lemma?

I speculate they’re the similar, possibly after stately additional constraints on the launch $U$. It vitality very nicely breathe that the relation is patent and simply reveals my need of judgement.

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