Isomorphism classes of line bundles over a scheme $X$ are described by the Picard group $Pic(X)$. Now there is a paper that describes the moduli space of line bundles with connections. This paper is quite technical for me to understand but I do understand that it must be quite different than $Pic(X)$. We can fix our $X$ (I am mainly interested in complex surfaces). Since we have fixed rank as well ($r=1$). I suspect that the moduli space I am interested in is a fibration over the Picard group (maybe of the moduli space of connections?).

Maybe we can take as an example $X = \mathbb{CP}^2$. Then we know that $Pic(X) = \mathbb{Z}$ since the invertible sheaves $=$ line bundles are classified by the degree of the corresponding divisor. Now, let us fix for a moment such a degree say $l$ for the class of line bundle associated to $\mathcal{O}(l)$. Equip this space with a connection $A \in \mathcal{A}^{(l)}$ where the latter is the affine space of connections over $X$. Of course we want to consider connections up to endomorphisms $f \in G^{(l)}$. Therefore, for the line bundle $\mathcal{O}(l)$ the corresponding moduli space of connections on it is $\mathcal{M}^{(l)} := \mathcal{A}^{(l)}/G^{(l}$. I can only assume then that the **moduli space of line bundles with connections** is a "combination" of $Pic(X)$ and $\sqcup_l \mathcal{M}^{(l)}$. How far off am I?

Of course I might be saying completely crazy things. I would like to get some intuition so anything you might say might be of some help.

P.S. In this question the OP claims that line bundles over $X$ are parametrised by their Chern classes. Is this parametrisation an isomorphism class one? And if so what is the relation with Pic($X$) and the question above?

Complex analytic connections in fibre bundles, Transactions of the Amer. Math. Soc. 85 (1957), 181-207. MR 19:172c. $\endgroup$ – Ben McKay Apr 2 '18 at 20:48