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Parallels between Moduli of Quiver Representations and Vector Bundles over Curves
Victoria Hoskins Freie Universität Berlin, Arnimallee 3, Raum 011, 14195 Berlin, Germany
Abstract:
This is a review article exploring similarities between moduli of quiver representations and moduli of vector bundles over a smooth projective curve. After describing the basic properties of these moduli problems and constructions of their moduli spaces via geometric invariant theory and symplectic reduction, we introduce their hyperkähler analogues: moduli spaces of representations of a doubled quiver satisfying certain relations imposed by a moment map and moduli spaces of Higgs bundles. Finally, we survey a surprising link between the counts of absolutely indecomposable objects over finite fields and the Betti cohomology of these (complex) hyperkähler moduli spaces due to work of Crawley-Boevey and Van den Bergh and Hausel, Letellier and Rodriguez-Villegas in the quiver setting, and work of Schiffmann in the bundle setting.
Keywords:
algebraic moduli problems; geometric invariant theory; representation theory of quivers; vector bundles and Higgs bundles on curves.
Received: September 25, 2018; in final form November 18, 2018; Published online December 4, 2018
Citation:
Victoria Hoskins, “Parallels between Moduli of Quiver Representations and Vector Bundles over Curves”, SIGMA, 14 (2018), 127, 46 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1426 https://www.mathnet.ru/eng/sigma/v14/p127
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Abstract page: | 182 | Full-text PDF : | 54 | References: | 17 |
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