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Moduli for decorated tuples of sheaves and representation spaces for quivers. (English) Zbl 1076.14019

In his previous work [Transform. Groups 9, No. 2, 167–209 (2004; Zbl 1092.14042)], the author gave a unified construction for many moduli spaces over curves. He introduced appropriate stability concepts and used geometric invariant theory to construct these coarse moduli spaces. In the paper under review, these results are extended in two directions: the dimension of the base manifold is now arbitrary and the group \(\text{GL}(r)\) which enters the definition of the objects to be classified, is replaced by a product of such groups. This corresponds to considering moduli problems which involve more than one vector bundle. The methods used to prove these generalisations are similar to those used in his previous work. Therefore, the author confines himself to highlight the main steps needed to adapt the proofs to the more general situation. Thus, large part of the effort goes into a clear and consistent formulation of the stability concepts, the definition of the moduli functors and the formulation of the main results. Apart from these fundamental results and as a motivation for this work, two applications to completely different moduli problems are presented. The first one deals with integral Gorenstein covers of degree four over a fixed projective manifold. The second one comes from representation theory of finite dimensional algebras. A stability concept for representations of quivers is introduced and the existence of a coarse moduli space is shown. The moduli space of Higgs bundles can be seen as a special case of this construction.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14H60 Vector bundles on curves and their moduli
16G20 Representations of quivers and partially ordered sets

Citations:

Zbl 1092.14042
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References:

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