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CURVATURE AND COMPLEX SINGULARITIES 483
We will first show that under the decomposition (4.23),
(4.24) (AV (R) AIT) @ (A"-W (R) A"-).
To verify this apply the automorphism z* pz to CN. The Maurer-Cartan
forms , (de,, e) and volume form on G(N- k, N) are preserved, while
(dz*, e,) po,. From (4.20) we ier that
which implies
On the other hand, if , p,,
from which we conclude that $ $,, thus establishing (4.24).
In terms of a frame {0; e, ., eN} lying over the fixed point and using the
index range (4.19), $ is a sum of terms
where P is a homogeneous polynomial of degree k and Q is one of degree n k.
When the frame undergoes a rotation
for unitary matrices g and h,
e ge, e, h,e
From the theory of unitary invariants it follows that is expressible in terms
of the quantities
(4.25)
co A o A 8 A , and
Of course we would like to show that is a constant multiple of (61 + 62) k A
det 1", which would prove (4.21). To establish (4.22) we observe that modulo
the differential ideal # {co,, dco,} the list (4.25) reduces to the quantities.