## Homogeneous supermanifolds associated with the complex projective line.(English)Zbl 0887.58003

Any Lie algebra of holomorphic vector fields defined on an open set of $$\mathbb{C}$$ can be globalized to the Lie algebra of fundamental vector fields corresponding to the standard action of $$SL_2 (\mathbb{C})$$ on $$\mathbb{C} P^1$$, to the standard action of the affine group on $$\mathbb{C}$$, or to the standard action of $$\mathbb{C}$$ on itself.
The authors consider a similar problem for Lie superalgebras of holomorphic vectors fields on the superspace $$\mathbb{C}^{1|m}$$, where $$m>0$$. The goal of the paper is to take a first step in this direction. Homogeneous supermanifolds (in the sense of Berezin-Leites) $$(M, {\mathcal O})$$ of dimension $$1\mid m$$ such that $$M= \mathbb{C} P^1$$ are studied. In the split case such supermanifolds are in a bijective correspondence with the nonincreasing $$m$$-tuples of nonnegative integers. The classification in the nonsplit case is not as simple and reduces to calculations with the tangent sheaf cohomology of split homogeneous supermanifolds. In the dimension $$1|2$$ there is only one nonsplit homogeneous supermanifold of the form $$(\mathbb{C} P^1, {\mathcal O})$$: it is the superquadric in $$\mathbb{C} P^{1|2}$$. When $$m=3$$, there is a series of nonsplit homogeneous supermanifolds. The classification of such supermanifolds when $$m\geq 4$$ is not yet completed.

### MSC:

 58A50 Supermanifolds and graded manifolds 58C50 Analysis on supermanifolds or graded manifolds
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### References:

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