View PDF - Project Euclid

CURVATURE AND COMPLEX SINGULARITIES 511
formula (5.30) can be established for a general 1-parameter family of metrics
which need not come from embeddings in IRN. Since any metric can be connected
by a linear homotopy to one coming from such an embedding, we have
thus established (5.29) for any metric. The point is that, just as the expressions
in Weyl’s tube formula turn out to be intrinsic invariants of the induced metric
and therefore have meaning for any Riemannian manifold, the same will be true
for the variation (5.30) of the Gauss-Bonnet integrand.
An interesting question is whether we can use a similar argument to establish
the topological invariance of the top Milnor number/x" + of a hypersurface
V0 c Cn / having an isolated singularity at the origin. This is especially intriguing
since the example of Briancon and Speder (cf. Tessier, loc. cit.) shows
that the lower Milnor numbers may not be topologically invariant. Suppose
then that {Vt,s} is a family of complex-analytic hypersurfaces parametrized by
h I where A is the disc {Itl < 1} and I is the real interval {0 _< --- s 1}, and
where Vt,s is smooth for 0 while V0, has an isolated singularity at the origin.
We assume that, for fixed s, the V,s fill out a neighborhood U of the origin, and
setting U* U- {0} define
3/: U* I (E" +1 X G(n, n + 1)
by y(z, s) (z, Tz(Vt,s)) where z Vt,s (t #-0). For any invariant curvature
polynomial Pk(IlE) on G(n, N) we set
on U* x I, so that
y*(b n-k A P(OE)) A ds
dp Vt,s 49"- /k P()vt,)"
Since this form is closed, as in (5.30) we deduce that
0
s dT
so that for # 0
(5.31)
0
( )Iv
pk(.Vt,s)/n_k=
t,s[ OV ,s[ e]
Observe that the right hand side of (5.31) is defined also for O.
If we now let be the form arising from c(O), then (5.23) together with
the topological invariance of the Milnor numbers suggests that
2(n k)
k 0 V0,s[e]
On the other hand, since the lower Milnor numbers are not topologically invariant,
we will not have
(5.33) lim
ez(,
O.