The punctured cotangent bundle (excluding the zero section) of the complex projective space of dimension $n$ can be identified with the space:
$$\{(P,Q) \mid P^2=P, tr(P)=1, PQ+QP = Q, \operatorname{tr}(Q) = 0, Q\ne 0\},$$
where $P$ and $Q$ are $(n+1)$ dimensional Hermitian matrices. Please see Furutani and Tanaka's article A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to a geometric quantization I (Proposition 2.3.).
Sketch of the proof: The space of the matrices $P$ is isomorphic as a homogeneous space to $CP^n$, since the action of $U(n+1)$ on it is transitive and the isotropy group of any point is $U(n)$. The conditions on the matrix $Q$ are obtained by taking the derivatives of the conditions on $P$, and the identification of $T^*CP^n$ and $TCP^n$ via the Fubini–Study metric.
