Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 072, 18 pages      arXiv:1508.02318      https://doi.org/10.3842/SIGMA.2016.072

Cohomology of the Moduli Space of Rank Two, Odd Degree Vector Bundles over a Real Curve

Thomas John Baird
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, A1C 5S7, Canada

Received October 30, 2015, in final form July 20, 2016; Published online July 22, 2016

Abstract
We consider the moduli space of rank two, odd degree, semi-stable Real vector bundles over a real curve, calculating the singular cohomology ring in odd and zero characteristic for most examples.

Key words: moduli space of vector bundles; gauge groups; real curves.

pdf (397 kb)   tex (23 kb)

References

  1. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
  2. Baird T.J., Moduli spaces of vector bundles over a real curve: $\mathbb{Z}/2$-Betti numbers, Canad. J. Math. 66 (2014), 961-992, arXiv:1207.4960.
  3. Baird T.J., Classifying spaces of twisted loop groups, Algebr. Geom. Topol. 16 (2016), 211-229, arXiv:1312.7450.
  4. Biswas I., Huisman J., Hurtubise J., The moduli space of stable vector bundles over a real algebraic curve, Math. Ann. 347 (2010), 201-233, arXiv:0901.3071.
  5. Crétois R., Real bundle automorphisms, in Cauchy-Riemann operators and orientability of moduli spaces, arXiv:1309.3110.
  6. Daskalopoulos G.D., The topology of the space of stable bundles on a compact Riemann surface, J. Differential Geom. 36 (1992), 699-746.
  7. Gross B.H., Harris J., Real algebraic curves, Ann. Sci. École Norm. Sup. (4) 14 (1981), 157-182.
  8. Harder G., Narasimhan M.S., On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975), 215-248.
  9. Kirwan F., The cohomology rings of moduli spaces of bundles over Riemann surfaces, J. Amer. Math. Soc. 5 (1992), 853-906.
  10. Liu C.-C.M., Schaffhauser F., The Yang-Mills equations over Klein surfaces, J. Topol. 6 (2013), 569-643, arXiv:1109.5164.
  11. McCleary J., A user's guide to spectral sequences, Cambridge Studies in Advanced Mathematics, Vol. 58, 2nd ed., Cambridge University Press, Cambridge, 2001.
  12. Okonek C., Teleman A., Abelian Yang-Mills theory on real tori and theta divisors of Klein surfaces, Comm. Math. Phys. 323 (2013), 813-858, arXiv:1011.1240.
  13. Schaffhauser F., Moduli spaces of vector bundles over a Klein surface, Geom. Dedicata 151 (2011), 187-206, arXiv:0912.0659.
  14. Schaffhauser F., Real points of coarse moduli schemes of vector bundles on a real algebraic curve, J. Symplectic Geom. 10 (2012), 503-534, arXiv:1003.5285.

Previous article  Next article   Contents of Volume 12 (2016)