= Cohomology ring of the orthogonal complex line flag manifold
{title2=$H^*(X_n;\mathbb Z)=\mathbb Z[x,y]/(x^n,\sum_{j=0}^{n-1}x^{n-1-j}y^j)$}
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 $x^iy^j$ with $0\leq i<n$, $0\leq j<n-1$. Polynomial division by the monic relation in $y$ proves that there are no further relations. Multiplying that relation by $y-x$ also gives $y^n=0$.
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