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The subbundles of decomposable vector bundles over an elliptic curve. (English) Zbl 0928.14022

From the introduction: In Proc. Lond. Math. Soc., III. Ser. 7, 414-452 (1957; Zbl 0084.17305), M. Atiyah classified the vector bundles over an elliptic curve \(C\). But still, there are several natural open question on the structure of the subbundles of a fixed vector bundle on \(C\). The aim of this paper is to give a reasonably complete answer to this question restricting slightly the vector bundles involved. A vector bundle \(F\) on a smooth projective curve is called polystable if it is the direct sum of stable vector bundles with the same slope \(\mu(F)\). In particular a polystable vector bundle is semistable. The notion of polystability is very natural over an elliptic curve, because very few vector bundles over an elliptic curve are stable, while for all integers \(r,d\) with \(r>0\) there exist polystable vector bundles with rank \(r\) and degree \(d\). This paper is devoted to the proof of the following result.
Theorem: Fix integers \(x,y,a_i\), \(1\leq i\leq x\), \(r_i\), \(1\leq i\leq x\), \(b_j\), \(1\leq j\leq y\), \(s_j\), \(1\leq j\leq y\), with \(x>0\), \(y>0\), \(r_i>0\) for every \(i\), \(s_j>0\) for every \(j\) and \(a_i/r_i< b_j/s_j\) for every \(i\) and every \(j\). Let \(C\) be an elliptic curve. Fix polystable vector bundles \(E_i\), \(1\leq i\leq x\), and \(F_j\), \(1\leq j\leq y\), with \(\text{rank} (E_i)=r_i\), \(\det(E_i) =a_i\), \(\text{rank} (F_j)=s_j\), \(\deg(F_j) =b_j\). Set \(E:= \bigoplus_{1 \leq i\leq x}E_i\), \(r:= \sum_{1\leq i\leq x}r_i=\text{rank}(E)\), \(F:= \bigoplus_{1\leq j\leq y}F_j\) and \(s:= \sum_{1\leq j\leq y}s_j= \text{rank}(F)\). We assume that no two among the indecomposable factors of \(E\) are isomorphic and that no two among the indecomposable factors of \(F\) are isomorphic. Then:
(a) If \(r\leq s\) there is an injective map \(f:E\to F\) and the general \(f\in H^0(C, \text{Hom} (E,F))\) has this property;
(b) If \(r<s\) there is an injective map \(f:E\to F\) with \(\text{Coker} (f)\) locally free and the general \(f\in H^0(C, \text{Hom} (E,F))\) has this property;
(c) If \(r>s\) there is a surjective map \(f:E\to F\) and the general \(f\in H^0(C, \text{Hom} (E,F))\) has this property.
A particular case of this theorem (part (b) with \(s=1)\) is the following result which was the main aim of this paper.
Corollary: Fix integers \(x\), \(a_i\), \(1\leq i\leq x\), \(r_i\), \(1\leq i\leq x\), \(b\) and \(s\) with \(x>0\), \(r_i>0\) for every \(i\), \(s>r:= \sum_{1\leq i\leq x}r_i\) and \(a_i/r_i<b/s\) for every \(i\). Let \(C\) be an elliptic curve. Fix polystable vector bundles \(E_i\), \(1\leq i\leq x\), and \(F\) with \(\text{rank}(E_i)= r_i\), \(\deg(E_i) =a_i\), \(\text{rank}(F)=s\), \(\deg(F)=b\). Set \(E:= \bigoplus_{1\leq i\leq x}E_i\). We assume that no two among the indecomposable factors of \(E\) are isomorphic and that no two among the indecomposable factors of \(R\) are isomorphic. Then \(F\) has a saturated subbundle isomorphic to \(E\).

MSC:

14H60 Vector bundles on curves and their moduli
14H52 Elliptic curves
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

Citations:

Zbl 0084.17305
Full Text: DOI EuDML

References:

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