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458 PHILLIP A. GRIFFITHS
For this reason, it will be convenient to use the manifold of all flames {Vl, V,
/
", Vn / z} for z. This is a complex manifold fibering holomorphically over
"
G(2, n + 2), and the structure equations
dvi Z Oo v.
dO 0 / 0
are valid as before, but where the {0} are now holomorphic differentials. In this
setting our problem is to determine the meaning of the conditions
for all , .
Ag 01/x/ 02t, 02g 01v 0
As observed above, the tri-linear algebra data relevant to analyzing the differential
f, is that of a linear map
A 2 Hom (2, n + 2/2).
Denote by (tl, t2) coordinates on 2 and think of A as being given by a
pencil A(t) Altl + A2t2 of 2 X n matrices. We may consider the 2 x 2 minors
of A(t) as giving
2 A(t) 22 2,
and we shall first assume that 2 A(t) O. For example, supposing that the
initial minor is non-zero we shall prove that A34 # 0. Effectively, we are then in
the case n 2 which we also assume in order to simplify notation.
Now 2 A(t) 2 2 22 is a quadratic function having two roots, which
we may take to be t 0 and t2 0 (the case of a double root must be treated
separately). Then A1 and A2 are both 2 x 2 matrices of rank one, and ff either
ker A1 ker A2 # 0, or
imA imA2#0
we deduce that 2 A(t) 0 for all t. It follows that we may choose a basis
Vl, v2, v3, v4 for 4 SO that v, v2 is a basis for 2 C 4 and
Then
and
AlV v3, Air2 0
A2v2 /)4, A2v -0.
013 dl, 024 dz2 023 014 0
A34 dz / dz 2.