Beschreibung:
It is well-known that any Lie supergroup G is split, i.e. its structure sheaf is isomorphic to the structure sheaf of a certain vector bundle. However, there are non-split complex homogeneous supermanifolds. We describe left invariant gradings of a complex homogeneous supermanifold G=H induced by gradings of the Lie supergroup G in terms of so called split grading operators. Sufficient conditions for a homogeneous supermanifold to be split are given in terms of Lie superalgebras and Lie subsuperalgebras.