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Mathematics > Algebraic Geometry

arXiv:1809.05738 (math)
[Submitted on 15 Sep 2018 (v1), last revised 30 Dec 2018 (this version, v2)]

Title:Parallels between moduli of quiver representations and vector bundles over curves

Authors:Victoria Hoskins
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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.
Subjects: Algebraic Geometry (math.AG)
Cite as: arXiv:1809.05738 [math.AG]
  (or arXiv:1809.05738v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1809.05738
Journal reference: SIGMA 14 (2018), 127, 46 pages
Related DOI: https://doi.org/10.3842/SIGMA.2018.127

Submission history

From: Victoria Hoskins [view email]
[v1] Sat, 15 Sep 2018 16:13:21 UTC (48 KB)
[v2] Sun, 30 Dec 2018 11:09:07 UTC (54 KB)
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