2017/06/05 by Sergio L. Cacciatori, Cacciatori, Sergio L., Simone Noja +3 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.1706.01354
openalex publication_date 2017/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℙ2 are completely characterised by a non-zero 1-form ω and by a locally free sheaf F of rank 0|2, satisfying Sym2 F ≅ Kℙ2. Denoting such supermanifolds with ℙ2ω(F), we show that all of them are Calabi-Yau supermanifolds and, when ω≠ 0, they are non-projective, that is they cannot be embedded into any projective superspace ℙn|m. Instead, we show that every non-projected supermanifolds over ℙ2 admits an embedding into a super Grassmannian. By contrast, we give an example of a supermanifold \mathbb P2ω(\mathcal F) that cannot be embedded in any of the Π-projective superspaces \mathbb PnΠ introduced by Manin and Deligne. However, we also show that when \mathcal F is the cotangent bundle over ℙ2, then the non-projected ℙ2ω(\mathcal F) and the Π-projective plane \mathbb P2Π do coincide.