Non projected calabi-yau supermanifolds over P2

We start a systematic study of non-projected supermanifolds, concentrating on supermanifolds with fermionic dimension 2 and with the reduced manifold a complex projective space. We show that all the non-projected supermanifolds of dimension 2j2 over P2 are completely characterised by a non-zero cohomology class 2 H1(TP2 (3)) and by a locally free sheaf FM of rank 0j2, satisfying Sym2FM = KP2 . Denoting such supermanifolds with P2 (FM ), we show that all of them are Calabi-Yau supermanifolds and, when 6= 0, they are non-projective, that is they cannot be embedded into any projective superspace Pnjm. Instead, we show that every non-projected supermanifold over P2 admits an embedding into a super Grassmannian. By contrast, we give an example of a supermanifold P2 (FM ) that cannot be embedded in any of the projective superspaces Pn introduced by Manin and Deligne. However, we also show that when FM is the cotangent bundle over P2, then the non-projected P2 (FM ) and the -projective plane P2 do coincide.