Homogeneous supermanifolds associated with complex projective space. I

Homogeneous and evenly homogeneous complex supermanifolds of dimension $n|m$ with reduction equal to the complex projective space $\mathbb {CP}^n$ are studied. Under the assumption that $m\leqslant n$, all split supermanifolds of this type are classified and their 1-cohomology with values in the tangent sheaf (which is invariant under the projective group) is calculated.