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, complex-analytic supermanifolds are not necessarily split, see for example the moduli space of super Riemann surfaces of genus $g>4$ (R.Donagi and E.Witten).

The talk is devoted to the splitting problem for complex homogeneous spaces. While any complex Lie supergroup G is split, there are examples of non-split complex homogeneous supermanifolds. In fact, almost all flag supermanifolds are non-split. The main question that we will discuss is how to determine if a given complex homogeneous supermanifold is split or non- split. Our results will be illustrated on super Grassmannians.