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

Abstract: 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.
 Comments: 17 pages. Exposition of the main theorem improved. Typos fixed Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph) Cite as: arXiv:1706.01354 [math.AG] (or arXiv:1706.01354v2 [math.AG] for this version)

## Submission history

From: Simone Noja [view email]
[v1] Mon, 5 Jun 2017 14:42:55 GMT (47kb)
[v2] Sun, 18 Mar 2018 17:25:52 GMT (29kb)