# Map from local systems to holomorphic line bundles on a curve

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.

• There is a typo., you want $Pic(X) = H^1(X,\mathcal{O}_X^*)$. One can show, with a bit of Hodge theory, that $H^1(X,U(1))$ (resp. $H^1(X,\mathbb{C}^*$)) maps bijectively (resp. surjectively) onto the Jacobian. I might add details later if I have time. Feb 29, 2020 at 22:34

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.