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.

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