Abstract
We obtain a classification up to isomorphism of complex analytic supermanifolds with underlying space \(\mathbb{CP}^1\) of dimension 1|2 and of dimension 1|3 with retract (k, k, k), where \(k\;\in\;\mathbb{Z}\). More precisely, we prove that classes of isomorphic complex analytic supermanifolds of dimension 1|3 with retract (k, k, k) are in one-to-one correspondence with points of the following set:
for k ≥ 2. For k > 2 all such supermanifolds are isomorphic to their retract (k, k, k). In addition, we show that classes of isomorphic complex analytic supermanifolds of dimension 1|2 with retract (k1, k2) are in one-to-one correspondence with points of \(\mathbb{CP}^{k_{1}+k_{2}-4}\;\mathrm{for}\;k_1+k_2\geq 5.\;\mathrm{For}\;k_1+k_2\;<\;5\) all such supermanifolds are isomorphic to their retract (k1, k2).
Mathematics Subject Classification (2010). 51P05, 53Z05, 32M10.
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© 2014 Springer International Publishing Switzerland
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Vishnyakova, E.G. (2014). On Complex Analytic 1|2- and 1|3-dimensional Supermanifolds Associated with \(\mathbb{CP}^1\). In: Kielanowski, P., Bieliavsky, P., Odesskii, A., Odzijewicz, A., Schlichenmaier, M., Voronov, T. (eds) Geometric Methods in Physics. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-06248-8_14
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DOI: https://doi.org/10.1007/978-3-319-06248-8_14
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-06247-1
Online ISBN: 978-3-319-06248-8
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