Non Projected Calabi-Yau Supermanifolds over $\mathbb{P}^2$

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 $2|2$ over $\mathbb{P}^2$ are completely characterised by a non-zero 1-form $\omega$ and by a locally free sheaf $\mathcal{F}$ of rank $0|2$, satisfying $Sym^2 \mathcal{F} \cong K_{\mathbb{P}^2}$. Denoting such supermanifolds with $\mathbb{P}^{2}_\omega(\mathcal{F})$, we show that all of them are Calabi-Yau supermanifolds and, when $\omega \neq 0$, they are non-projective, that is they cannot be embedded into any projective superspace $\mathbb{P}^{n|m}$. Instead, we show that every non-projected supermanifolds over $\mathbb{P}^2$ admits an embedding into a super Grassmannian. By contrast, we give an example of a supermanifold $\mathbb P^{2}_\omega(\mathcal F)$ that cannot be embedded in any of the $\Pi$-projective superspaces $\mathbb P^{n}_{\Pi}$ introduced by Manin and Deligne. However, we also show that when $\mathcal F$ is the cotangent bundle over $\mathbb{P}^2$, then the non-projected $\mathbb{P}^2_\omega(\mathcal F)$ and the $\Pi$-projective plane $\mathbb P^{2}_{\Pi}$ do coincide.