Advertisement

On Complex Analytic 1|2- and 1|3-dimensional Supermanifolds Associated with \(\mathbb{CP}^1\)

  • E. G. VishnyakovaEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

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:
$${\bf{Gr}}_{4k-4,3}\cup{\bf{Gr}}_{4k-4,2}\cup{\bf{Gr}}_{4k-4,1}\cup{\bf{Gr}}_{4k-4,0}$$
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).

Keywords

Complex analytic supermanifold projective line 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of LuxembourgLuxembourgLuxembourg

Personalised recommendations