Let $\mathcal{H}\subset{\mathbb{C}}{\mathbb{P}}^{n}$ be the union of the $n+2$ generic hyperplanes in ${\mathbb{C}}{\mathbb{P}}^{n}$. Let $\mathcal{U}\subset{\mathbb{C}}{\mathbb{P}}^{n}$ be the union of their $\epsilon$-neighborhoods for a very small $\epsilon>0$.

The complement $\bar{\mathcal{P}}_{n}={\mathbb{C}}{\mathbb{P}}^{n}\smallsetminus\mathcal{U}$ is a manifold with boundary and corners. We call $\bar{\mathcal{P}}_{n}$ the $n$-dimensional pair-of-pants. We call $\mathcal{P}_{n}={\mathbb{C}}{\mathbb{P}}^{n}\smallsetminus\mathcal{H}$ the $n$-dimensional open pair-of-pants