Supergeometry of Π-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 ΩPjavax.xml.bind.JAXBElement@2499360f1. 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,ΩPjavax.xml.bind.JAXBElement@14aca00a1,λ), where Pn is the ordinary complex projective space, ΩPjavax.xml.bind.JAXBElement@1bab69961 is its cotangent sheaf and λ is a non-zero complex number, representative of the fundamental obstruction class ω∈H1(TPjavax.xml.bind.JAXBElement@53a5c4eb⊗⋀2ΩPjavax.xml.bind.JAXBElement@3ee1b9de1)≅C. Likewise, in the case n=1 the Π-projective line PΠ1 is the split supermanifold determined by the pair (P1,ΩPjavax.xml.bind.JAXBElement@4f3cf2f81≅OPjavax.xml.bind.JAXBElement@f9bd550(−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.