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510 PHILLIP A. GRIFFITHS was done in section l(b), where KdM is the Gauss-Kronecker curvature. b) Using a) and the tube construction, deduce (5.29) for submanifolds mn C IR u aS was also done in section l(b), this step explains the origin of the Gauss-Bonnet integrand. c) Following the discussion in l(c), use the Gauss mapping y M-- GR(n, N) to show that Cm KdM y*Pf(l).) is induced from a closed form on the Grassmannian. d) Finally, to show that the left hand side of (5.29) is independent of the Riemannian metric g on M, we proceed as follows: Denote by I {0 -< s
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.
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we deduce that CURVATURE AND COMPLE
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so that CURVATURE AND COMPLEX SINGU
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