= Flag manifold
{title2=$F_{d_1,\ldots,d_m}(V)$}
{wiki}
For $0<d_1<\cdots<d_m<n$, a flag is a chain of complex subspaces $L_1\subset\cdots\subset L_m\subset V$ of the given dimensions. Successively choosing the quotient subspaces gives complex dimension $\sum_{i=1}^m(d_i-d_{i-1})(n-d_i)$ with $d_0=0$. Adapted bases identify it with a homogeneous space of the <special linear group>, with stabilizer an appropriate block upper-triangular subgroup. The <twistor correspondence> uses $F_{1,2}(\mathbb C^4)$.
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