Abstract
An odd deformation of a super Riemann surface \(\mathcal {S}\) is a deformation of \(\mathcal {S}\) by variables of odd parity. In this article, we study the obstruction theory of these odd deformations \(\mathcal {X}\) of \(\mathcal {S}\). We view \(\mathcal {X}\) here as a complex supermanifold in its own right. Our objective in this article is to show, when \(\mathcal {X}\) is a deformation of second order of \(\mathcal {S}\) with genus \(g>1\): if the primary obstruction class to splitting \(\mathcal {X}\) vanishes, then \(\mathcal {X}\) is in fact split. This result leads naturally to a conjectural characterisation of odd deformations of \(\mathcal {S}\) of any order.
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Notes
In fact, a stronger result is obtained: that a holomorphic projection of \(\mathfrak {M}_g\) to its reduced space is obstructed. This implies \(\mathfrak {M}_g\) is itself non-split.
However, as pointed out by Donagi and Witten [7, pp. 29–30], it is a more subtle issue to construct deformations of non-split superspaces.
If we were in a more general setting where the object under deformation had (even) dimension greater than one, the bracket \([\psi _1, \psi _2]\) on the right-hand side of (3.21) would represent an obstruction to the existence of a deformation.
The limit over \(\mathfrak {U}\) is taken over common refinement on the underlying space \(\mathcal {X}_\mathrm {red}\). Indeed, for the kinds of supermanifolds considered in this article, their topology is concentrated in their reduced part.
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Acknowledgements
The author would like to firstly acknowledge the Australian National University where much of this work was undertaken; the anonymous referee for useful suggestions on the improvement of this article; and finally the Yau Mathematical Sciences Centre, Tsinghua University, where support was given while preparing revisions of this article.
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Bettadapura, K. On the problem of splitting deformations of super Riemann surfaces. Lett Math Phys 109, 381–402 (2019). https://doi.org/10.1007/s11005-018-1112-x
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DOI: https://doi.org/10.1007/s11005-018-1112-x