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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 = 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 S^{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 = 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

    \[F \cong P_{( r^{\prime}, d^{\prime})} \otimes S^{c-1} V \otimes L,\]

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

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