On the Structure of Complex Homogeneous Supermanifolds

Details On the Structure of Complex Homogeneous Supermanifolds On the Structure of Complex Homogeneous Supermanifolds

2008-11-16

arxiv.org/abs/0811.2581v1

Mathematics

Differential Geometry

12 pages

Scientific paper

For a Lie group $G$ and a closed Lie subgroup $H\subset G$, it is well known that the coset space $G/H$ can be equipped with the structure of a manifold homogeneous under $G$ and that any $G$-homogeneous manifold is isomorphic to one of this kind. An interesting problem is to find an analogue of this result in the case of supermanifolds. In the classical setting, $G$ is a real or a complex Lie group and $G/H$ is a real and, respectively, a complex manifold. Now, if $G$ is a real Lie supergroup and $H\subset G$ is a closed Lie subsupergroup, there is a natural way to consider $G/H$ as a supermanfold. Furthermore, any $G$-homogeneous real supermanifold can be obtained in this way, see \cite{Kostant}. The goal of this paper is to give a proof of this result in the complex case.

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