Regularity of Gauss Manin connection

Question

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.

Answer 1

$H_{\mathbb C}$ 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.

You can take a look on pp5-6 of Brian Conrad "CLASSICAL MOTIVATION FOR THE RIEMANN–HILBERT CORRESPONDENCE"