= Orthogonal complex line flag manifold
{title2=$X_n=\{(\ell_1,\ell_2):\ell_1\perp\ell_2\}\cong\mathbb P(L^\perp)$}
For $n\geq2$, ordered orthogonal complex lines in $\mathbb C^n$ form a <projective bundle> over $\mathbb{CP}^{n-1}$, with fiber $\mathbb{CP}^{n-2}$. Here $L$ is the first tautological line and the complement is formed using the standard <Hermitian inner product>. Its real dimension is $4n-6$. It is also the partial complex flag space of a line contained in a two-plane: send an orthogonal pair to $(\ell_1,\ell_1\oplus\ell_2)$, and recover the second line by orthogonal complement inside the two-plane.
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