  We review nonabelian Poisson structures on affine and projective spaces over
$\mathbb{C}$. We also construct a class of examples of nonabelian Poisson
structures on $\mathbb{C} P^{n-1}$ for $n>2$. These nonabelian Poisson
structures depend on a modular parameter $\tau\in\mathbb{C}$ and an additional
descrete parameter $k\in\mathbb{Z}$, where $1\leq k<n$ and $k,n$ are coprime.
The abelianization of these Poisson structures can be lifted to the quadratic
elliptic Poisson algebras $q_{n,k}(\tau)$.
