${\mathbb{H}}_{2n+1}$-structures on odd dimensional projective spaces

We prove that the Heisenberg group ${\text{h}}_{2n+1}$ admits infinitely many inequivalent equivariant compactifications into ${\mathbb{P}}^{2n+1}$ for all $n\ge 1$. This result provides an analog of Hassett-Tschinkel’s classical result beyond commutative algebraic groups.