Let $$X = \mathbb{C} / \Lambda$$, where $$\Lambda$$ is the lattice generated by $$1, \tau \in \mathbb{C}$$, for some $$\tau$$ with $$\text{Im}(\tau) > 0$$. We want to show some explicit forms for the moduli space of line bundles over $$X$$, the so called Picard group $$\text{Pic}(X)$$. This moduli space is endowed with a group structure provided by tensor products, this post intends to give a brief exposition on the structure of the Picard groups and the Chern classes of elliptic curves.

Let $$\mathcal{O}_X$$ be the sheaf of holomorphic functions on $$X$$, and let $$\mathcal{O}_X^*$$ be the sheaf of invertible holormorphic functions. We have the following.

Proposition: The Picard group is isomorphic to the Čech cohomology group $$H^1(X, \mathcal{O}_X^*)$$.

Proof: Let $$(t_{\alpha, \beta}, U_\alpha \cap U_\beta)$$ be an element of $$H^1(X, \mathcal{O}_X^*)$$ where $$U_\alpha$$ is an open cover of $$X$$, by definition it satisfies

$1 = dt_{\alpha\beta\gamma} = t_{\alpha\beta} t_{\beta\gamma} t_{\gamma\alpha}$

on $$U_\alpha \cap U_\beta \cap U_\gamma$$. Consider $$L = \left(\bigcup_\alpha U_\alpha \times \mathbb{C} \right) /\sim$$, where if $$x \in U_\alpha \cap U_\beta$$, $$(x, u)_\alpha \sim (x, t_{\beta \alpha} u)_\beta$$. One can check that thanks to the co-cycle identity this defines a equivalence relation and we have natural charts $$U_\alpha \times \mathbb{C} \to L$$ given by $$t_\alpha$$ which define a holomorphic line bundle structure in $$L$$. On the other hand if $$t_{\alpha\beta}$$ is trivial, i.e. there are functions $$g_\alpha \in \mathcal{O}(U_\alpha) ^*$$ such that $$t_{\alpha\beta} = g_\alpha/g_\beta$$, then . Conversely, fixing any trivialization $$\varphi_\alpha$$, noting that the maps $$\varphi_\alpha \varphi_\beta^{-1}$$ are given by an element of $$t_{\alpha\beta}: U_\alpha \cap U_\beta \to GL(1, \mathbb{C}) = \mathbb{C}^*$$, we have that $$t_{\alpha\beta} t_{\beta\gamma} t_{\gamma\alpha}$$ is given by $$\varphi_\alpha\varphi_\beta^{-1}\varphi_\beta \varphi_\gamma^{-1}\varphi_\gamma\varphi_\alpha^{-1} = Id$$ so the co-cycle identity holds.

Let $$\mathbb{Z}$$ be the sheaf of locally constant $$\mathbb{Z}$$-valued functions on $$X$$. Consider the exact sequence

$0 \longrightarrow \mathbb{Z}_X \longrightarrow \mathcal{O}_X \overset{exp}{\longrightarrow} \mathcal{O}_X^* \longrightarrow 0.$

This induces an exact sequence

$H^1(X,\mathbb{Z}_X)\longrightarrow H^1(X, \mathcal{O}_X) \longrightarrow H^1(X, \mathcal{O}_X^*)\overset{c_1}{\longrightarrow} H^2(X,\mathbb{Z}_X)$

## The Chern class of a line bundle

We define the Chern class of a line bundle $$L$$ as $$c_1(L) \in H^2(X,\mathbb{Z})$$. Everything in the previous diagram is natural so for any morphism $$f:X \to Y$$ we have that $$f^*c_1(L) = c_1(f^*L)$$ for any line bundle $$L \in \text{Pic}(Y)$$.

Proposition: A complex line bundle $$L$$ is topologically trivial if and only if $$c_1(L) = 0$$.

Proof: Let $$C^\infty_X$$ be the sheaf of smooth functions on $$X$$. Now we have

$H^k(X,C^\infty_X) = 0, \qquad k > 0.$

So repeating the exercise on the induced long exact sequence for

$0 \longrightarrow \mathbb{Z}_X \longrightarrow C^\infty_X \longrightarrow {C^\infty_X}^* \longrightarrow 0.$

We get that $$c_1: H^1(X, {C^\infty_X}^*) \to H^2(X, \mathbb{Z})$$ is an isomorphism. The natural morphism $$\mathcal{O}_X \to C^\infty_X$$ induces a map of the previosly mentioned long exact sequences that it is the identity on $$\mathbb{Z}_X$$, this means that $$c_1(L) = 0$$ holds as a complex line bundle if and only if it holds as a smooth line bundle as a smooth line bundle and since $$c_1$$ is an isomorphism for smooth line bundles, $$L$$ is a topologically trivial line bundle. $$\square$$

By Dolbeaut’s theorem and the fact that the de Rham cohomology satisfies $$H^p(X,\mathbb{Z}) = H^p(X,\mathbb{Z}_X)$$, the long exact sequence is reduced to

$0 \longrightarrow H^1(X,\mathbb{Z})\longrightarrow H^{0,1}(X, \mathbb{C}) \longrightarrow \text{Pic}(X)\overset{c_1}{\longrightarrow} H^2(X,\mathbb{Z}) \longrightarrow 0$

So we identify the kernel $$\text{Pic}_0(X)$$ of $$c_1$$ with the quotient $$H^{0,1}(X, \mathbb{C}) / H^1(X,\mathbb{Z})$$. Now, as

$h^{0,1}(X,\mathbb{C}) = h^{0,1}(X,\mathbb{C}), \quad h^1(X,\mathbb{C}) = 2.$

The space $$H^{0,1}(X, \mathbb{C})$$ can be identified with $$\mathbb{C} \cdot d\bar{z} \cong \overline{\mathbb{C}}^*$$ and since $$H_1(X, \mathbb{Z})$$ can be identified with $$\Lambda$$, via Poincaré duality we can see that $$H^1(X,\mathbb{Z})$$ is given by

$\Lambda^* = \{ u \mid \langle u, \lambda \rangle \in \mathbb{Z},\quad \forall\ \lambda \in \Lambda \}.$

We now have an identification $$\text{Pic}_0(X)=\overline{\mathbb{C}}/\Lambda^*$$, this endows $$\text{Pic}_0(X)$$ with the structure of an elliptic curve. Note that $$X$$ is isomorphic to $$\text{Pic}_0(X)$$ and $$\text{Pic}(X)/ \text{Pic}_0(X) = H^2(X,\mathbb{Z})$$.

## Bibliography

1. Griffiths, P. Principles of algebraic geometry. Wiley. 1978.

2. Miranda, R. Algebraic curves and Riemann surfaces. American Mathematical Soc. 1995.