= Projective-space quotient as a Thom space of conjugate tautological lines
{title2=$\mathbb{CP}^{n+m+1}/\mathbb{CP}^{n}\cong\operatorname{Th}(\bar\gamma^{\oplus(n+1)}\to\mathbb{CP}^{m})$}
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 $\gamma^*\otimes V$, 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
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