Connections on vector bundles over elliptic curves - concrete computations

Question

This is linked to my question on math.Stackexchange for which I had no answer.

I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me fully grasp this object.(except maybe Katz's paper) that links the notion of connection on a vector bundle to:

solvability of homogeneous systems of first-order differential equations(a question first raised by Grothendieck).

Let $X$ be an elliptic curve and let $(U_i)_i$ be an open covering of $X$.

A connection on a vector bundle $\nabla|_{U_i} = d_{X} + A_i$ where $A_i$ is a $2\times 2$ matrix with entries $1$-form in $\Omega^1_X$.

1. How can we make sense of $\nabla|_{U_i}$, in other words, how can we add a scalar ($\Omega^1$ is of dimension $1$ as $X$ is smooth) with a matrix?
2. I would be happy if you have any references on this subject.

Mnay thanks for your help.

Answer 1

Perhaps the following very elementary discussion will help you to understand the relationship between systems of first order ODE's, connections, and vector bundles. If what I write below isn't helpful, I apologize for wasting your time.

Let $u: \mathbb{C}\rightarrow \mathbb{C}^{n}$ be a vector valued holomorphic function. Let $A: \mathbb{C}\rightarrow M_{n}(\mathbb{C})$ be a holomorphic function valued in $n \times n$ matrices.

Consider the differential equation

$\frac{\partial}{\partial z} u(z)= A(z)u(z),$

where the juxtaposition on the right hand side is matrix multiplication.

This is a system of first order holomorphic differential equations. This equation can be solved using fundamental theorems in ODE, but it's not so important for the discussion.

Now, consider the trivial holomorphic vector bundle $\mathbb{C}\times \mathbb{C}^{n}$ over $\mathbb{C}.$ Then, the vector valued holomorphic function $u: \mathbb{C}\rightarrow \mathbb{C}^{n}$ becomes a section of this vector bundle.

Furthermore, the above mentioned system of first order differential equations can be viewed as a (holomorphic) connection $D_{A}$ on this bundle.

To wit, given a (local) holomorphic section $u,$ we declare that

$D_{A}u=\frac{\partial}{\partial z} u(z) \ dz-A(z)u(z)\ dz.$

In particular, $D_{A}u=0$ if and only if it solves the aforementioned ODE.

So, we see that $D_{A}$ sends a section of the trivial bundle to a holomorphic $1$-form with values in sections of the trivial bundle. Checking the Leibniz rule is straightforward, and hence $D_{A}$ is a connection.

In order words, a (holomorphic) connection on the trivial bundle $\mathbb{C}\times \mathbb{C}^{n}$ over $\mathbb{C}$ is just a first order system of homogeneous holomorphic differential equations. The general $1$-dimensional complex manifold is not biholomorphic to $\mathbb{C},$ and therefore the trivial bundle becomes a rank $n$-vector bundle, and the first order system of differential equations becomes a "local system." It's a shame, in my opinion, that this reason for the terminology is not more well known, though it is certainly known to experts.

Deriving the transformation law you state for a connection is now a reasonably simple matter of making sense of how coordinate changes of a nontrivial vector bundle act on the relevant ODE that is specifying a connection.

I admit, I haven't even begun to answer your specific question, but I hope this explanation isn't totally useless.