Let $X$ be a Riemann surface of genus $g > 0$. Let $S$ denote the set of local systems (locally constant sheaves) on $X$ with fiber $\mathbb{C}$. $S$ is in natural bijection with $H^1(X, \underline{\mathbb{C}^{\times}})) \cong (\mathbb{C}^{\times})^{2g}$, where $\underline{\mathbb{C}^{\times}}$ is the constant sheaf on $X$ with fiber $\mathbb{C}^{\times}$. 


By the Riemann-Hilbert correspondence, each local system $\mathbb{L} \in S$ corresponds to a line bundle with a flat connection. This map is given by $\mathbb{L} \mapsto (\mathbb{L} \otimes\_{\mathbb{C}} \mathcal{O}\_X, \nabla = 1 \otimes d)$, where $d$ is the usual exterior derivative. Forgetting the connection gives a map from $S$ to the Picard group of $X$, $H^1(X, \mathcal{O}\_X)$. This is the natural map $H^1(X, \underline{\mathbb{C}}^{\times}) \to H^1(X, \mathcal{O}\_X^{\*})$, so in particular it is a map of abelian groups.


What is the description of this map? In particular, I do not know how to show it has non-trivial image, i.e., that there is a non-trivial holomorphic line bundle that admits a flat connection.


It follows from Chern-Weil theory that the image is contained in the Jacobian (as the curvature of a connection is essentially the first Chern class). The identification of $S$ with $(\mathbb{C}^{\times})^{2g}$ is non-canonical, but the topology this gives to $S$ is canonical. I think one can show that this map is continuous by thinking about cocycles, which gives another proof that the image is in the Jacobian. 



Map from local systems to holomorphic line bundles on a curve
I think the following theorem answers your question.



> 
> **Theorem:** Let $X$ be a smooth, proper connected curve over $\mathbf C$ with a line bundle $\mathscr L$. Then $\mathscr L$ admits a flat connection $\nabla$ if and only if $c\_1(\mathcal L)=0$. 
> 
> 
> 


**Remark:** A more general statement would be that a line bundle $\mathscr L$ on a compact Kahler manifold admits an integrable connection if and only if $c\_1(\mathscr L)\otimes \mathbf Q=0$.


Before we go to the proof of this theorem, we recall that *define* the first Chern class as the connecting homomorphism 
$$
c\_1\colon \mathrm{Pic}\_X=\mathrm{H}^1(X,\mathscr O\_X^\*) \to \mathrm{H}^1(X,\mathbf Z(1))
$$
that comes from the exponential short exact sequence
$$
0 \to \mathbf Z(1) \to \mathscr O\_X \xrightarrow{exp(-)} \mathscr O\_X^\* \to 0 \ .
$$


**Proof:** The essential idea is to relate the exponential sequences for $\mathbf C$ and $\mathscr O\_X$ with each other. More precisely, we have a morhpism of short exact sequences:
$$0 \to \mathbf Z(1) \to \mathscr O\_X \to \mathscr O\_X^\* \to 0 \\
\downarrow \\
0 \to \mathbf Z(1) \to \underline{\mathbf C} \to \underline{\mathbf C}^\* \to 0 \ .
$$
So we have a commutative diagram


$$
\mathrm{H}^1(X,\mathbf C) \to \mathrm{H}^1(X,\mathbf C^\*) \xrightarrow{\delta} \mathrm{H}^2(X,\mathbf Z(1)) \\ 
\downarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \downarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \downarrow{\mathrm{Id}} \\ 
\mathrm{H}^1(X,\mathscr O\_X) \to \mathrm{H}^1(X,\mathscr O\_X^\*) \xrightarrow{c\_1} \mathrm{H}^2(X,\mathbf Z(1) \ .
$$


**Step 1.** The map $\delta$ is the zero map. 


Probably the easiest way is to say that $\mathrm{H}^1(X,\mathbf C^\*)=(\mathbf C^\*)^{2g}$ and $\mathrm{H}^2(X,\mathbf Z(1))=\mathbf Z$. But there are no non-trivial homomorphisms from $(\mathbf C^\*)^{2g}$ to $\mathbf Z$.


**Step 2.** The map $c\_1(\mathscr L)=0$ if $\mathscr L$ admits a flat connection. 


Riemann-Hilbert correspondence says that $\mathscr L$ admits a flat connection if and only if it lies in the image of $\mathrm{H}^1(X,\mathbf C^\*) \to \mathrm{H}^1(X,\mathscr O\_X^\*)$. Combining it with Step 1, we conclude the claim. 


**Step 3.** If $c\_1(\mathscr L)=0$ then $\mathscr L$ is in the image of $\mathrm{H}^1(X,\mathbf C^\*) \to \mathrm{H}^1(X,\mathscr O\_X^\*)$.


This easily follows from the commutative diagram above and the fact that $\mathrm{H}^1(X,\mathbf C) \to \mathrm{H}^1(X,\mathscr O\_X)$ is surjective. The latter fact is, in turn, a consequence of degeneration of the Hodge-to-de Rham spectral sequence 
$$
\mathrm{E}^{p,q}\_2=\mathrm{H}^q(X, \Omega^p\_X) \Rightarrow \mathrm{H}^{p+q}(X, \mathbf C) \ .
$$ 


**Step 4.** If $c\_1(\mathscr L)=0$ then $\mathscr L$ admits a flat connection.


Again, this is just a consequence of Step 3 and Riemann-Hilbert Correspondence.


