On complex-analytic $1|3$-dimensional supermanifolds associated with $\mathbb{CP}^1$

Abstract

We obtain a classification up to isomorphism of complex-analytic supermanifolds with underlying space $\mathbb{CP}^1$ of dimension $1|3$ with retract $(k,k,k)$, where $k\in \mathbb{Z}$. More precisely, we prove that classes of isomorphic complex-analytic supermanifolds of dimension $1|3$ with retract $(k,k,k)$ are in one-to-one correspondence with points of the following set: $\mathbf{Gr}_{4k-4,3} \cup \mathbf{Gr}_{4k-4,2} \cup \mathbf{Gr}_{4k-4,1} \cup \mathbf{Gr}_{4k-4,0}$ for $k \geq 2$. For $k < 2$ all such supermanifolds are isomorphic to their retract $(k,k,k)$.