Projective-space quotient as a Thom space of conjugate tautological lines (${\mathbb{CP}}^{n+m+1}/{\mathbb{CP}}^n\cong \mathrm{Th}({\bar{\gamma }}^{\oplus (n+1)}\to {\mathbb{CP}}^m)$)

Decompose the ambient vector space as $V\oplus W$, of dimensions $n+1,m+1$. A projective line outside $\mathbb{P}(V)$ is the graph of a unique linear map from a line in $W$ to $V$. Thus the complement is the total space of ${\gamma }^{\ast }\otimes V$, which is complex-isomorphic to ${\bar{\gamma }}^{\oplus (n+1)}$ after choosing a Hermitian metric. Collapsing $\mathbb{P}(V)$ gives its one-point compactification, hence the Thom space because the base is compact.

// Target: geometry-and-topology.bigb