Supergeometry of Î -projective spaces

In this paper we prove that Π-projective spaces PΠn arise naturally in supergeometry upon considering a non-projected thickening of Pn related to the cotangent sheaf ΩPn1. In particular, we prove that for n⩾2 the Π-projective space PΠn can be constructed as the non-projected supermanifold determined by three elements (Pn,ΩPn1,λ), where Pn is the ordinary complex projective space, ΩPn1 is its cotangent sheaf and λ is a non-zero complex number, representative of the fundamental obstruction class ω∈H1(TPn⊗⋀2ΩPn1)≅C. Likewise, in the case n=1 the Π-projective line PΠ1 is the split supermanifold determined by the pair (P1,ΩP11≅OP1(−2)). Moreover we show that in any dimension Π-projective spaces are Calabi-Yau supermanifolds. To conclude, we offer pieces of evidence that, more in general, also Π-Grassmannians can be constructed the same way using the cotangent sheaf of their underlying reduced Grassmannians, provided that also higher, possibly fermionic, obstruction classes are taken into account. This suggests that this unexpected connection with the cotangent sheaf is characteristic of Π-geometry.