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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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