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

SIGMA 14 (2018), 094, 12 pages      arXiv:1804.06366      https://doi.org/10.3842/SIGMA.2018.094

Higher Obstructions of Complex Supermanifolds

Kowshik Bettadapura
Yau Mathematical Sciences Center, Tsinghua University, Haidian, Beijing, 100084, China

Received April 29, 2018, in final form August 30, 2018; Published online September 07, 2018

Abstract
In this article we introduce the notion of a `good model' in order to study the higher obstructions of complex supermanifolds. We identify necessary and sufficient conditions for such models to exist. Illustrations over Riemann surfaces are provided.

Key words: complex supergeometry; supermanifolds; obstruction theory.

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References

  1. Bartocci C., Bruzzo U., Hernández Ruipérez D., The geometry of supermanifolds, Mathematics and its Applications, Vol. 71, Kluwer Academic Publishers Group, Dordrecht, 1991.
  2. Berezin F.A., Introduction to superanalysis, Mathematical Physics and Applied Mathematics, Vol. 9, D. Reidel Publishing Co., Dordrecht, 1987.
  3. Bettadapura K., On the problem of splitting deformations of super Riemann surfaces, Lett. Math. Phys., to appear, arXiv:1610.07541.
  4. Brylinski J.-L., Loop spaces, characteristic classes and geometric quantization, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008.
  5. Deligne P., Morgan J.W., Notes on supersymmetry (following Joseph Bernstein), in Quantum Fields and Strings: a Course for Mathematicians, Vols. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 1999, 41-97.
  6. Donagi R., Witten E., Super Atiyah classes and obstructions to splitting of supermoduli space, Pure Appl. Math. Q. 9 (2013), 739-788, arXiv:1404.6257.
  7. Donagi R., Witten E., Supermoduli space is not projected, in String-Math 2012, Proc. Sympos. Pure Math., Vol. 90, Amer. Math. Soc., Providence, RI, 2015, 19-71, arXiv:1304.7798.
  8. Green P., On holomorphic graded manifolds, Proc. Amer. Math. Soc. 85 (1982), 587-590.
  9. Grothendieck A., A general theory of fibre spaces with structure sheaf, University of Kansas, 1995.
  10. Manin Yu.I., Gauge field theory and complex geometry, Grundlehren der Mathematischen Wissenschaften, Vol. 289, Springer-Verlag, Berlin, 1988.
  11. Okonek C., Schneider M., Spindler H., Vector bundles on complex projective spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2011.
  12. Onishchik A.L., On the classification of complex analytic supermanifolds, Lobachevskii J. Math. 4 (1999), 47-70.
  13. Palamodov V.P., Invariants of analytic $Z_{2}$-manifolds, Funct. Anal. Appl. 17 (1983), 68-69.

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