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 $2\mathrm{\mid }2$ over ${\mathbb{P}}^2$ are completely characterised by a non-zero 1-form $\omega$ and by a locally free sheaf $\mathcal{F}$ of rank $0\mathrm{\mid }2$, satisfying $Sy{m}^2\mathcal{F}\cong K_{{\mathbb{P}}^2}$. Denoting such supermanifolds with ${\mathbb{P}}_{\omega }^2(\mathcal{F})$, we show that all of them are Calabi-Yau supermanifolds and, when $\omega \mathrm{\ne }0$, they are non-projective, that is they cannot be embedded into any projective superspace ${\mathbb{P}}^{n\mathrm{\mid }m}$. 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 ${\mathbb{P}}_{\omega }^2(\mathcal{F})$ that cannot be embedded in any of the $\mathrm{\Pi }$-projective superspaces $\mathbb P^{n}_{\Pi}$introducedbyManinandDeligne\mathrm{.}However,wealsoshowthatwhen$\mathcal F$isthecotangentbundleover$$\mathbb{P}$^2$, then the non-projected ${\mathbb{P}}_{\omega }^2(\mathcal{F})$ and the $\mathrm{\Pi }$-projective plane ${\mathbb{P}}_{\mathrm{\Pi }}^2$ do coincide.