d_{0} = 0

and assume

0 < d_{1} < \dots < d_{m} < n

. Choose the first member of the flag manifold as a

d_{1}

-plane, contributing

d_{1} (n - d_{1})

L_{i - 1}

, choose

L_{i} / L_{i - 1}

(d_{i} - d_{i - 1})

V / L_{i - 1}

(d_{i} - d_{i - 1}) (n - d_{i})

. Adding these fiber dimensions gives

dim_{C} F_{d_{1}, \dots, d_{m}} (V) = i = 1 \sum m (d_{i} - d_{i - 1}) (n - d_{i}) .

For

V = C^{4}

the results are

dim_{C} F_{1} = 3, dim_{C} F_{2} = 4, dim_{C} F_{12} = 5.

Here

F_{1} = CP^{3}

F_{2} = Gr (2, 4)

F_{12}

is the twistor correspondence space of incident pairs consisting of a spacetime point and a twistor on its twistor line. Over

F_{2}

its fiber is

CP^{1}

; over

F_{1}

its fiber is

CP^{2}

, consistent with

4 + 1 = 3 + 2 = 5

. These are complex dimensions; the underlying real dimensions are

6, 8, 10

, while a real spacetime slice of

F_{2}

has real dimension four.

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