Flag manifold ($F_{d_1,\dots ,d_m}(V)$)

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 $F_{1,2}({\mathbb{C}}^4)$.