Parallels between Moduli of Quiver Representations and Vector Bundles over Curves
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...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут математики НАН України
2018
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209877 |
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| Zitieren: | Parallels between Moduli of Quiver Representations and Vector Bundles over Curves / V. Hoskins // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 65 назв. — англ. |
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Hoskins, V. 2025-11-28T09:39:09Z 2018 Parallels between Moduli of Quiver Representations and Vector Bundles over Curves / V. Hoskins // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 65 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14D20; 14L24; 16G20; 14H60 arXiv: 1809.05738 https://nasplib.isofts.kiev.ua/handle/123456789/209877 https://doi.org/10.3842/SIGMA.2018.127 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. This article is based on lecture notes for the fifth workshop on the Geometry and Physics of Higgs bundles, and the author would like to thank Laura Schaposnik for the organisation of this workshop. The author would also like to thank the participants of a seminar on counting indecomposable quiver representations held at the Freie Universität Berlin for interesting discussions related to this topic. The author is supported by the Excellence Initiative of the DFG at the Freie Universität Berlin. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Parallels between Moduli of Quiver Representations and Vector Bundles over Curves Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Parallels between Moduli of Quiver Representations and Vector Bundles over Curves |
| spellingShingle |
Parallels between Moduli of Quiver Representations and Vector Bundles over Curves Hoskins, V. |
| title_short |
Parallels between Moduli of Quiver Representations and Vector Bundles over Curves |
| title_full |
Parallels between Moduli of Quiver Representations and Vector Bundles over Curves |
| title_fullStr |
Parallels between Moduli of Quiver Representations and Vector Bundles over Curves |
| title_full_unstemmed |
Parallels between Moduli of Quiver Representations and Vector Bundles over Curves |
| title_sort |
parallels between moduli of quiver representations and vector bundles over curves |
| author |
Hoskins, V. |
| author_facet |
Hoskins, V. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209877 |
| citation_txt |
Parallels between Moduli of Quiver Representations and Vector Bundles over Curves / V. Hoskins // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 65 назв. — англ. |
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AT hoskinsv parallelsbetweenmoduliofquiverrepresentationsandvectorbundlesovercurves |
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2025-12-07T18:18:45Z |
| last_indexed |
2025-12-07T18:18:45Z |
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1850886113071202304 |