On complex analytic 1|2- and 1|3-dimensional supermanifolds associated with ℂℙ1

Abstract

We obtain a classification up to isomorphism of complex analytic supermanifolds with underlying space ℂℙ1 of dimension 1|2 and of dimension 1|3 with retract (k, k, k), where k ∈ 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: Gr4k-4,3 ⊂ Gr4k-4,2 ⊂ Gr4k-4,1 ⊂ Gr4k-4,0 for k ≥ 2. For k < 2 all such supermanifolds are isomorphic to their retract (k, k, k). In addition, we show that classes of isomorphic complex analytic supermanifolds of dimension 1|2 with retract (k1, k2) are in one-to-one correspondence with points of ℂℙk1+k2-4 for k1 + k2 ≥ 5. For k1 + k2 < 5 all such supermanifolds are isomorphic to their retract (k1, k2).