  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.
