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Moduli problems of sheaves associated with oriented trees. (English) Zbl 1033.14009

Let \((X,H)\) be a polarized variety. The author studies the following problem: Given an oriented tree \(Q\), we want to have a moduli space of vector bundles for each vertex and morphism of vector bundles for each edge of the tree \(Q\). These vector bundles and morphisms must have predescribed numerical invariants. The main result is theorem 1.6 which states the existence of a coarse moduli space if we are restricted to \(\theta\)-stable bundles and \(S\)-equivalence classes of \(\theta\)-semistable ones. Here the concept of \(\theta\)-semistability arises naturally from the GIT construction which is carried out in this article. The easy tree \(Q= ( \bullet \rightarrow \bullet)\) leads to what is called holomorphic triples. It turns out that for holomorphic triples \(\theta\)-stability generalizes the \(\tau\)-stability of Bradlow and García-Prada. The tree \(Q\) is used to build inductively large trees.

MSC:

14D22 Fine and coarse moduli spaces
16G20 Representations of quivers and partially ordered sets
14D20 Algebraic moduli problems, moduli of vector bundles
14L24 Geometric invariant theory
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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