# Supergeometry of $\Pi$-Projective Spaces

## TL;DR

This paper demonstrates that $ ext{Pi}$-projective spaces naturally emerge in supergeometry as non-projected supermanifolds linked to the cotangent sheaf, revealing their Calabi-Yau nature and potential generalizations to $ ext{Pi}$-Grassmannians.

## Contribution

It introduces a novel construction of $ ext{Pi}$-projective spaces using the cotangent sheaf and obstruction classes, highlighting their geometric and Calabi-Yau properties.

## Key findings

- $ ext{Pi}$-projective spaces are non-projected supermanifolds derived from the cotangent sheaf.
- For $n
eq 1$, $ ext{Pi}$-projective spaces are Calabi-Yau supermanifolds.
- Evidence suggests $ ext{Pi}$-Grassmannians can be constructed similarly, involving higher obstruction classes.

## Abstract

In this paper we prove that $\Pi$-projective spaces $\mathbb{P}^n_\Pi$ arise naturally in supergeometry upon considering a non-projected thickening of $\mathbb{P}^n$ related to the cotangent sheaf $\Omega^1_{\mathbb{P}^n}$. In particular, we prove that for $n \geq 2$ the $\Pi$-projective space $\mathbb{P}^n_\Pi$ can be constructed as the non-projected supermanifold determined by three elements $(\mathbb{P}^n, \Omega^1_{\mathbb{P}^n}, \lambda)$, where $\mathbb{P}^n$ is the ordinary complex projective space, $\Omega^1_{\mathbb{P}^n}$ is its cotangent sheaf and $\lambda $ is a non-zero complex number, representative of the fundamental obstruction class $\omega \in H^1 (\mathcal{T}_{\mathbb{P}^n} \otimes \bigwedge^2 \Omega^1_{\mathbb{P}^n}) \cong \mathbb{C}.$ Likewise, in the case $n=1$ the $\Pi$-projective line $\mathbb{P}^1_\Pi$ is the split supermanifold determined by the pair $(\mathbb{P}^1, \Omega^1_{\mathbb{P}^1} \cong \mathcal{O}_{\mathbb{P}^1} (-2)).$ Moreover we show that in any dimension $\Pi$-projective spaces are Calabi-Yau supermanifolds. To conclude, we offer pieces of evidence that, more in general, also $\Pi$-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 $\Pi$-geometry.