An explicit construction of indecomposable vector bundles over an elliptic curve

In the celebrated paper “Vector bundles over an elliptic curve,” M. Atiyah classifies indecomposable vector bundles, namely he provides a bijection $\alpha_{(r, d)}$ between indecomposable bundles of arbitrary rank $r$ and degree $d$ (denoted by $\mathcal{E}_{(r, d)}$ ) and $\mathcal{E}_{(h, 0)}$ (where $h = \text{gcd}(d, r)$ ). The latter is described explicitly: there is a distinguished element $F_r \in \mathcal{E}_{(r, 0)}$ such that for any other $N \in \mathcal{E}_{(r, 0)}$ one has $N \cong F_r \otimes L$ with $L$ a line bundle of degree $0$ . This $F_r$ is the unique element (up to isomorphism) in $\mathcal{E}_{(r, 0)}$ with nontrivial global sections. Also $F_r \cong \operatorname{Sym}^{r-1} V$ , where $V$ is a nontrivial extension $0 \rightarrow \mathcal{O} \rightarrow V \rightarrow \mathcal{O} \rightarrow 0$ .

However, since the construction of $\alpha_{(r, d)}$ is done in many steps, there is no explicit description of elements of $\mathcal{E}_{(r, d)}$ in the paper. I would like to share a construction, which was explained to me by Vadim Vologodsky.

Let $F$ be an indecomposable vector bundle of a rank $r$ and degree $d$ on an elliptic curve $E$ . Let $c = \text{gcd}(r, d)$ , $r = r^{\prime} c$ , $d = d^{\prime} c$ . Let $\pi : E \rightarrow E$ be a covering of degree $r^{\prime}$ and Galois group $\mathbb{Z}/r^{\prime}\mathbb{Z}$ . Denote $P_{( r^{\prime}, d^{\prime})} = \pi_* (\mathcal{O}(d^{\prime}))$ . Then

$F \cong P_{( r^{\prime}, d^{\prime})} \otimes \operatorname{Sym}^{c-1} V \otimes L,$

for $L \in \operatorname{Pic}^0(E)/\operatorname{Pic}^0(E)[ r^{\prime}].$

Let us prove the statement by induction. First assume that $r$ and $d$ are coprime. Then since $\deg(\pi^*(F) \otimes \mathcal{O}(-d)) = 0$ there exists a line sub bundle $L$ of $\pi^*(F) \otimes \mathcal{O}(-d)$ of degree $0$ (this is because of the equivalence between the category of semi-stable bundles of the slope $0$ and the category of torsion sheaves on $E$ ). Then one has $\pi_*(L(d)) \cong F$ (since $\pi_*(L(d))$ is stable) and we get the result.

Now assume that $c > 1$ . Since the degree of $\pi^*(F)$ is $r^{\prime} d$ we see that $\pi^*(F) \otimes \mathcal{O}(-\frac{r^{\prime}d}{ r})$ is a semi-stable bundle of degree $0$ . As before, there exists a line sub bundle $L_1$ of degree $0$ . Thus we have $L_1(\frac{r^{\prime}d}{ r}) \rightarrow \pi^*(F)$ , and thus $\pi_*(L_1(\frac{r^{\prime}d}{ r})) \rightarrow F$ . Thus we have $0 \rightarrow P_{( r^{\prime}, d^{\prime})} \otimes L \rightarrow F$ for a line bundle $L$ on $E$ with $\pi^*(L) \cong L_1$ .

The cokernel is indecomposable since $F$ is indecomposable, so by induction it is isomorphic to $P_{( r^{\prime}, d^{\prime})} \otimes \operatorname{Sym}^{c-2}V \otimes N$ where $N$ is a line bundle of degree $0$ . Actually, one can see that if $L \neq N$ then $\text{Ext}^1$ is trivial, and since $F$ is indecomposable it can’t be the case. Now it suffices to check that $\text{Ext}^1(P_{( r^{\prime}, d^{\prime})} \otimes \operatorname{Sym}^{t}V, P_{( r^{\prime}, d^{\prime})})$ is one dimensional. But it is easy to prove by induction that it is isomorphic to $\text{Ext}^1(\operatorname{Sym}^{t}V, \mathcal{O})$ , which is one-dimensional.