Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-25T23:05:24.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Moduli Spaces of Vector Bundles over a Real Curve: ℤ/2-Betti Numbers

Published online by Cambridge University Press:  20 November 2018

Thomas Baird
Thomas Baird*
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, A1C 5S7. e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute $\mathbb{Z}/2$–Betti numbers of these spaces.

Keywords

32L0514P25cohomology of moduli spacesholomorphic vector bundles
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 5 , 01 October 2014 , pp. 961 - 992
Copyright
Copyright © Canadian Mathematical Society 2014

References

[AB83] Atiyah, M. F. and Bott, R., The Yang–Mills equations over Riemann surfaces. Philos.. Trans. Roy. Soc. London Ser. A 308(1983), 523–615 http://dx.doi.org/10.1098/rsta.1983.0017.Google Scholar
[BHH10] Biswas, I., Huisman, J., and Hurtubise, J. C., The moduli space of stable vector bundles over a real algebraic curve. Math. Ann. 347(2010), 201–233.http://dx.doi.org/10.1007/s00208-009-0442-5 Google Scholar
[Dol63] Dold, A., Partitions of unity in the theory of fibrations. Ann. of Math. 78(1963), 223–255. http://dx.doi.org/10.2307/1970341 Google Scholar
[GH81] Gross, B. H. and Harris, J., Real algebraic curves. Ann. Sci. École Norm. Sup. (4) 14(1981), 157–182.Google Scholar
[GH04] Goldin, R. F. and Holm, T. S., Real loci of symplectic reductions. Trans. Amer. Math. Soc. 356(2004), 4623–4642.http://dx.doi.org/10.1090/S0002-9947-04-03504-4 Google Scholar
[Gro57] Grothendieck, A., Sur la classification des fibrés holomorphes sur la sphere de Riemann. Amer. J. Math. 79(1957), 121–138.http://dx.doi.org/10.2307/2372388 Google Scholar
[HN75] Harder, G. and Narasimhan, M. S.,On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann. 212(1975), 215–248.http://dx.doi.org/10.1007/BF01357141 Google Scholar
[LS13] Liu, C. C. M. and Schaffhauser, F., Yang–Mills equations over Klein surfaces.. arxiv:1109.5164v3 (2013). http://dx.doi.org/10.1112/jtopol/jtt001 Google Scholar
[McC01] McCleary, J., A user’s guide to spectral sequences. Cambridge University Press, Cambridge, 2001.Google Scholar
[Mil56a] Milnor, J., Construction of universal bundles, I. Ann. of Math. 63(1956), 272–284.http://dx.doi.org/10.2307/1969609 Google Scholar
[Mil56b] Milnor, J., Construction of universal bundles, II. Ann. of Math. 63(1956), 430–436.http://dx.doi.org/10.2307/1970012 Google Scholar
[MM65] Milnor, J.W. and C. Moore, J., On the structure of Hopf algebras. Ann. of Math. 81(1965), 211–264.http://dx.doi.org/10.2307/1970615 Google Scholar
[MS74] Milnor, J.W. and Stasheff, J. D., Characteristic Classes. Princeton University Press, Princeton, NJ, 1974.Google Scholar
[Mum62] Mumford, D., Projective invariants of projective structures and applications. In: Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag–Leffler, Djursholm, 1963, 526–530.Google Scholar
[PS60] Palais, R. and Stewart, T., Deformations of compact differentiable transformation groups. Amer. J. Math. 82(1960), 935–937.http://dx.doi.org/10.2307/2372950 Google Scholar
[Sch11] Schaffhauser, F., Moduli spaces of vector bundles over a Klein surface. Geom. Dedicata 151(2011), 187–206.http://dx.doi.org/10.1007/s10711-010-9526-3 Google Scholar
[Sch12] Schaffhauser, F. Real points of coarse moduli schemes of vector bundles on a real algebraic curve. J. Symplectic Geom. 10(2012), 503–534.http://dx.doi.org/10.4310/JSG.2012.v10.n4.a2 Google Scholar
[Smi70] Smith, L., Lectures on the Eilenberg–Moore spectral sequence. Springer-Verlag, 1970.Google Scholar
[SW10] Saveliev, N. and Wang, S., On real moduli spaces of holomorphic bundles over M-curves. Topology Appl. 158(2011), 344–351.http://dx.doi.org/10.1016/j.topol.2010.11.005 Google Scholar