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CURVATURE AND COMPLEX SINGULARITIES 459
The case of a double root is similar to an argument we are about to give, and
will therefore be omitted.
Now suppose that/2 A(t) O. Then for all t, the linear transformation
A(t) (2 (n + 2/(I2
has a kernel containing a non-zero vector v(t). If the line (v(t) is not constant in
t, then we may choose Vl, v2 so that Alv2 0 A2vl. Then the vectors Alvl and
A2v2 are proportional modulo Vl, v2 since otherwise we would not have
/V A(t) O. It follows that we may choose a frame {vl, v2, ", vn / .} so that
AlVl 1)3, A2V2 v3, and AlV2 A2Vl O.
Then 013 dzl, 023 dz2, and all other 0., 0; thus
A33 2013 / 023 2 dgl/ dg2
is non-zero.
In case v(t) vl is constant in t, we will have the two cases"
Alv2 and A2v2 independent modulo vl, v2
Alv2 proportional to A2v2 modulo vl, v2.
In the first case we may choose a local frame so that
where 023/ 024
dr2
0. From
dr1
021V -1- - - 011Vl -" 012v2
022V2 023V3 024/)4
0 dO 13 012 / 023, 0 dO14 012 / 024
we deduce that 012 0. Then dr1
--
=-0 modulo Vl, which says exactly that the
line vi C z is constant in z. Finally, in the
--
second case we may choose a
local frame so that
dvi 011v1 012v2
dr2 021/)1 + 022v2
from which it follows that the rank of f, is one and so f(S) is a curve in
023v3,
G(2, n + 2). Q.E.D.
We will apply (3.7) to the holomorphic Gauss mapping. Suppose that
M C s is a complex manifold and denote by if(M) the manifold of Darboux
frames {z; el, ", eN} defined by the conditions
M; el," ", en give a unitary basis for Tz(M); and en + 1,
z. ", eN give a unitary basis for the normal space Nz(M).
We may think of (M) C (s) as an integral manifold of the differential system
to, 0, and then (3.1) and (3.2) become