# Regularity of Gauss Manin connection

I want to understand the "Regularity of Gauss Manin connection" from the most basic example. Suppose we have a family of projective manifold $$X\rightarrow \mathbb C^*$$ with full rank, then for any kth cohomology group, we have holomorphic bundle $$H_{\mathbb C}$$ with filtration $$F^p$$ coupled with the Gauss Manin connection $$\nabla: F^P\rightarrow F^{p-1}\otimes\Omega^1.$$ I have a some couple of questions here.

1 I saw one definition of Regularity in Griffiths's paper, which means in terms of a local rational basis of $$H_{\mathbb C}$$ (holomorphic on $$\mathbb C^*$$), the connection form has poles at the origion.

I feel confused about the existence of rational sections which can form a basis of $$H_{\mathbb C}$$. Why they exists. Also I little bit confused about the terminology "rational". Since it is required to be holomorphic outside the origion, why not just call it holomorphic section.

2 Is there any concrete example computed to see this regularity in this context. Reference will be helpful.

• If you can read french, Deligne's "Equations différentielles à points singuliers réguliers" (LNM 163) is certainly the best reference.
– abx
Commented Jun 16, 2022 at 9:54
• There is also a translation available at labs.thosgood.com/translations/978-3-540-05190-9.pdf Commented Jun 16, 2022 at 13:07
• @abx@pbelmans Thanks. The translation will be very helpful. Commented Jun 16, 2022 at 16:28

$$H_{\mathbb C}$$ is a holomorphic vector bundle over a punctured disk, so it is trivial since every holomorphic vector bundle over $$\mathbb C^*$$ is trivial, hence, there exist global holomorphic sections $$\eta_1,...,\eta_r$$ over $$\mathbb C^*$$ which are trivialization of $$H_{\mathbb C}$$.
I guess "rational" here because construction of $$H_{\mathbb C}$$ is purely algebraic (it is analytification of algebraic vector bundle associated with locally free sheaf $$R^k f_* \mathbb C \otimes_{\mathbb C} \mathcal O_{\mathbb C^*}$$), hence $$\eta_i$$ could be chosen from algebraic sections of $$H_{\mathbb C}$$ over $$\mathbb C^*$$ which are rational sections over $$\mathbb C$$, but more context is required to say for sure.
• @xinfu This is a part of the definition, not a statement. "We say $D$ have regular singular points on infinity if for any compactification $\overline{S}$, for any point $\overline{s} \in \overline{S} - S$ there exist topological neighborhood $U$ of $\overline{s}$ and raitonal sections $\eta_1,...,\eta_r$ of $E$ over $S$ which are holomorphic basis of $E$ over $U \cap S$ such that..." Thus, if vector bundle $E$ is not holomorphically trivializable near some point $\overline{s} \in \overline{S} - S$ (which could be the case) then $D$ is not a connection with regular singular points on infinity. Commented Jun 18, 2022 at 8:47