Symmetric theta divisors of Klein surfaces

Christian Okonek; Andrei Teleman

Open Mathematics (2012)

  • Volume: 10, Issue: 4, page 1314-1320
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Christian Okonek, and Andrei Teleman. "Symmetric theta divisors of Klein surfaces." Open Mathematics 10.4 (2012): 1314-1320. <http://eudml.org/doc/269734>.

@article{ChristianOkonek2012,
abstract = {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.},
author = {Christian Okonek, Andrei Teleman},
journal = {Open Mathematics},
keywords = {Klein surfaces; Symmetric theta divisors; Appell-Humbert data; Yang-Mills connections; Real vector bundles; symmetric theta divisors; real vector bundles},
language = {eng},
number = {4},
pages = {1314-1320},
title = {Symmetric theta divisors of Klein surfaces},
url = {http://eudml.org/doc/269734},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Christian Okonek
AU - Andrei Teleman
TI - Symmetric theta divisors of Klein surfaces
JO - Open Mathematics
PY - 2012
VL - 10
IS - 4
SP - 1314
EP - 1320
AB - 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.
LA - eng
KW - Klein surfaces; Symmetric theta divisors; Appell-Humbert data; Yang-Mills connections; Real vector bundles; symmetric theta divisors; real vector bundles
UR - http://eudml.org/doc/269734
ER -

References

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  9. [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 Zbl1037.57025
  10. [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. [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 Zbl1218.32007
  12. [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 Zbl0853.53022

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