Homogeneous supermanifolds associated with the complex projective line (Q1356242)

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.