On the splitting problem for complex supermanifolds

A supermanifold is called split if it is isomorphic to a vector bundle with a purely even base and purely odd fiber. In the smooth category, all supermanifolds are known to be split (although non-canonically). In contrast with that, complex-analytic supermanifolds are not necessarily split.

It is well-known that any complex Lie supergroup $G$ is split. However, there are non-split complex homogeneous supermanifolds. For instance, almost all flag supermanifolds are non-split. The main question that we shall discuss is how to find out whether a given complex supermanifold is split or non-split.