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Locally free sheaves on complex supermanifolds

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Abstract

A classification of locally free sheaves \( \mathcal{E} \) of \( \mathcal{O} \)-modules which have a given retract gr\( \mathcal{E} \) in the terms of non-abelian 1-cohomology is given. In the case of \( \mathbb{C}{{\mathbb{P}}^{1|m }} \), m > 0, we show that the Birkhoff–Grothendieck Theorem does not hold true. We obtain a result similar to the Barth–Van de Ven–Tyurin Theorem for projective superspaces. Furthermore, a spectral sequence which connects the cohomology with values in a locally free sheaf \( \mathcal{E} \) to the cohomology with values in its retract gr\( \mathcal{E} \) is constructed.

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Authors and Affiliations

  1. Department of Mathematics, Yaroslavl University, Yaroslavl, 150000, Russia

    A. L. Onishchik

  2. Max–Planck–Institut für Mathematik, 53072, Bonn, Germany

    E. G. Vishnyakova

Authors
  1. A. L. Onishchik
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  2. E. G. Vishnyakova
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Correspondence to A. L. Onishchik.

Additional information

1Supported by SFB TR|12 and by RFBR (grant no. 11-01-00465).

2Supported by MPI Bonn, SFB TR|12, MFO and by RFBR (grant no. 11-01-00465).

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Onishchik, A.L., Vishnyakova, E.G. Locally free sheaves on complex supermanifolds. Transformation Groups 18, 483–505 (2013). https://doi.org/10.1007/s00031-013-9219-8

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