The weighted projective superspace with weights $+1, -1$ and an analog of the Fubini--Study form

As a by-product of our work on super Pl\"{u}cker embedding, we came to the notion of a weighted projective superspace P+1,-1(V⊕ W) with weights $+1,-1$. The construction is not in itself super and makes sense in ordinary (purely even) framework. Unlike the familiar weighted projective spaces with positive weights, the (super)space P+1,-1(V⊕ W) is a smooth (super)manifold. We describe its structure and show that it possesses an analog of the Fubini--Study form.