The Picard group of an elliptic variety
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 cocycle 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 cocycle 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

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

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