About the lemma in proving the classical Riemann-Hilbert correspondence I was trying to understand the following lemma given in Brain Conrad's note about  Classical motivation for the Riemann Hilbert correspondence. The lemma goes as follows:

Lemma 1.6. If $X$ is smooth and $(\mathscr{E}, \nabla)$ is a vector
bundle on $X$ with a connection, then for the
$\underline{\mathbf{C}}$-module $\operatorname{ker} \nabla$ on $X$ the
$\mathbf{C}$-linear map $(\operatorname{ker} \nabla)_x \rightarrow
 \mathscr{E}(x)$ is injective for all $x \in X$. In particular, all
stalks of ker $\nabla$ are finite-dimensional over $\mathbf{C}$.
The sets $X \leq d=\left\{x \in X \mid
 \operatorname{dim}(\operatorname{ker} \nabla)_x \leq d\right\}$ are a
locally finite collection of closed sets in $X$, and ker $\nabla$ has
locally constant restriction to $X^{\leq d}-X \leq(d-1)$ for all d. In
particular, ker $\nabla$ is locally constant on $X$ if and only if $x
 \mapsto \operatorname{dim}(\operatorname{ker} \nabla)_x$ is a locally
constant function on $X$.


Proof. We may work locally on $X$, so we can assume $X=\mathbf{B}^n$ is a polydisc and $\mathscr{E}$ is globally free. Let $z_1, \ldots, z_n$ be the standard coordinates on $\mathbf{B}^n$, and let $\left\{e_j\right\}$ be a basis of $\mathscr{E}$. The sections $\mathrm{d} z_i \otimes e_k$ give a basis of $\Omega_X^1 \otimes \mathscr{E}$, and so if we expand $\nabla\left(e_j\right)$ with respect to this basis we see that the general condition $\nabla\left(\sum s_i e_i\right)=0$ is a system of first-order linear PDE's in the $s_i$ 's with coefficients expressed in terms of the $\Gamma_{i j}^k$ 's as in Example 1.2. The uniqueness theorem for ODE's with an initial condition may be applied to restrictions along analytic curves to conclude that a section of ker $\nabla$ over a connected open $U$ is uniquely determined by the specification of its value at a single point $x_0$. Thus, $(\operatorname{ker} \nabla)_x \rightarrow \mathscr{E}(x)$ is indeed injective for all $x$ and the sets $X-X^{\leq d}$ are open (so each $X^{\leq d}$ is closed). It is likewise clear that the collection of closed sets $\left\{X^{\leq d}\right\}_d$ is locally finite on $X$.
Pick $x \in X^{\leq d}-X^{\leq(d-1)}$, so $d=\operatorname{dim}(\operatorname{ker} \nabla)_x$. By shrinking $X$ around $x$ we may arrange that there is a $d$-dimensional subspace $V \subseteq(\operatorname{ker} \nabla)(X)$ with $V$ mapping isomorphically onto $(\operatorname{ker} \nabla)_x$. It follows that $\mathscr{O}_X \otimes_{\underline{\mathbf{C}}} \underline{V} \rightarrow \mathscr{E}$ is a map of vector bundles on the manifold $X$ such that it induces an injection on $x$-fibers. By shrinking $X$ around $x$, we may therefore arrange that this map is a subbundle, and hence the induced map $V=\underline{V}_{x^{\prime}} \rightarrow \mathscr{E}\left(x^{\prime}\right)$ is injective for all $x^{\prime} \in X$
......

There are several statements in the proof that I can't work out:

*

*Firstly, Why "there is a $d$-dimensional subspace $V \subseteq(\operatorname{ker} \nabla)(X)$ with $V$ mapping isomorphically onto $(\operatorname{ker} \nabla)_x$. It follows that $\mathscr{O}_X \otimes_{\underline{\mathbf{C}}} \underline{V} \rightarrow \mathscr{E}$ is a map of vector bundles on the manifold $X$ such that it induces an injection on $x$-fibers."   in the second paragraph. Help Please.

*Secondly, why "It is likewise clear that the collection of closed sets $\left\{X^{\leq d}\right\}_d$ is locally finite on $X$"

 A: I recently read his note too. Here is my attempt towards the sentences you point out.

*

*The key point is what does "shrinking $X$ around $x$" means. I believe he means to take a neighborhood of $x$ with given property but hard to say. Denote $\widetilde{X}$ for the space after shrinking $X$, required to satisfies:
($X$ is not the polydisc $\mathbf{B}^{n}$ here)



*

*$\widetilde{X}$ is an open neighborhood containing $x$;

*$(\text{ker}\nabla)(\widetilde{X}) $ is a $\underline{\mathbf{C}} $-module, and of dimension $\geq d$ as a vector space over $\underline{\mathbf{C}} $.


The existence of such $\widetilde{X}$ comes from: using definition of algebraic vector bundle locally get a $\underline{\mathbf{C}} $-module $\mathscr{E}(U)$ around $x$, and regard $(\text{ker}\nabla)(U) $ as $\underline{\mathbf{C}} $-submodule of $\mathscr{E}(U) $. Then apply the same ODE result that $X^{>d-1}$ is open. Take the intersection of $U$ and $X^{>d-1}$, named $\widetilde{X}$.
The rest in this question is easy to check.

*

*I think in the first paragraph, things are based on classical ODE Existence and Uniqueness Theorem plus analytic continuation. See [Wolfgang Walter, Ordinary Differential Equations] Section 21, Theorem II. However, I didn't understand the difference between $G$ $simple \ connected$ there and Brian's "connected U". Hope you are a master of ODE and able to explain that.

