Orthogonal complex line flag manifold ($X_n=\{({\ell }_1,{\ell }_2):{\ell }_1\perp {\ell }_2\}\cong \mathbb{P}(L^{\perp })$)

For $n\ge 2$, 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.

For the orthogonal complex line flag manifold, take $x,y$ to be the first Chern classes of the duals of its two tautological lines, both in degree two. The Whitney sum formula for Chern classes gives $c_i(L^{\perp })=x^i$. The projective bundle definition of Chern classes gives the second relation, and Leray-Hirsch theorem gives the integral basis $x^i{y}^j$ with $0\le i<n$, $0\le j<n-1$. Polynomial division by the monic relation in $y$ proves that there are no further relations. Multiplying that relation by $y-x$ also gives $y^n=0$.