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Quivers, vector bundles and coverings of smooth curves. (English) Zbl 1117.14033
A quiver bundle \(E\) on a projective variety is a set of vector bundles \(\{E_{v}\}_{v\in V}\) one for each vertex \(v \in V\) a finite quiver \(Q\), together with morphisms \(E_{v} \to E_{u}\), one for each arrow in \(Q\) from \(v \in V\) to \(u \in V\). Accordingly, one can define a notion of slope-stability of \(E\). Now consider a finite map \(f : X \to Y\), where \(Y\) is a smooth connected projective curve over a field of characteristic zero, and let \(E\) be a quiver bundle on \(Y\). Then the main result of the paper asserts that \(E\) is semistable (polystable) if and only if \(f^{*}(E)\) is semistable (polystable). The second part of the paper is devoted to the construction of examples of stable quiver bundles on curves which are non-étale double covers of a smooth elliptic curve. In this part the quiver under consideration should take the form of a multiple arrow, a sink, an oriented chain, or a fork.
MSC:
14H60 Vector bundles on curves and their moduli
16G20 Representations of quivers and partially ordered sets
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