# 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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