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2012 | 10 | 4 | 1314-1320
Tytuł artykułu

Symmetric theta divisors of Klein surfaces

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EN
Abstrakty
EN
This is a slightly expanded version of the talk given by the first author at the conference Instantons in complex geometry, at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces. In this paper we compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
4
Strony
1314-1320
Opis fizyczny
Daty
wydano
2012-08-01
online
2012-05-31
Twórcy
Bibliografia
  • [1] Arbarello E., Cornalba M., Griffiths Ph.A., Harris J., Geometry of Algebraic Curves. I, Grundlehren Math. Wiss., 267, Springer, New York, 1985
  • [2] Atiyah M.F., Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup., 1971, 4(1), 47–62
  • [3] Birkenhake Ch., Lange H., Complex Abelian Varieties, 2nd ed., Grundlehren Math. Wiss., 302, Springer, Berlin, 2004
  • [4] Costa A.F., Natanzon S.M., Poincaré’s theorem for the modular group of real Riemann surfaces, Differential Geom. Appl., 2009, 27(5), 680–690 http://dx.doi.org/10.1016/j.difgeo.2009.03.008
  • [5] Gross B.H., Harris J., Real algebraic curves, Ann. Sci. École Norm. Sup., 1981, 14(2), 157–182
  • [6] Ho N.-K., Liu C.-C.M., Yang-Mills connections on nonorientable surfaces, Comm. Anal. Geom., 2008, 16(3), 617–679
  • [7] Johnson D., Spin structures and quadratic forms on surfaces, J. London Math. Soc., 1980, 22(2), 365–373 http://dx.doi.org/10.1112/jlms/s2-22.2.365
  • [8] Libgober A., Theta characteristics on singular curves, spin structures and Rohlin theorem, Ann. Sci. École Norm. Sup., 1988, 21(4), 623–635
  • [9] Okonek Ch., Teleman A., Gauge theoretical equivariant Gromov-Witten invariants and the full Seiberg-Witten invariants of ruled surfaces, Comm. Math. Phys., 2002, 227(3), 551–585 http://dx.doi.org/10.1007/s002200200637
  • [10] Okonek Ch., Teleman A., Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces, preprint available at http://arxiv.org/abs/1011.1240
  • [11] Schaffhauser F., Moduli spaces of vector bundles over a Klein surface, Geom. Dedicata, 2011, 151, 187–206 http://dx.doi.org/10.1007/s10711-010-9526-3
  • [12] Wang S., A Narasimhan-Seshadri-Donaldson correspondence over non-orientable surfaces, Forum Math., 1996, 8(4), 461–474 http://dx.doi.org/10.1515/form.1996.8.461
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0048-0
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