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

TL;DR

This paper classifies complex-analytic supermanifolds of dimension 1|3 over p^1 with a specific retract, establishing a correspondence with points in certain Grassmannian varieties for ^1, and showing triviality for smaller retracts.

Contribution

It provides a complete classification of 1|3-dimensional supermanifolds over p^1 with a given retract, linking isomorphism classes to Grassmannian points.

Findings

Classifies supermanifolds with retract (k,k,k) for k 

Establishes correspondence with Grassmannians ^{4k-4}

Shows trivial supermanifolds for k < 2

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 $(k,k,k)$.