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432 PHILLIP A. GRIFFITHS
top Milnor number/.,(n + 1)(V0) a topological invariant of the isolated singularity,
and arrive at the formula
(O. lO) /.t(n+l)(Vo)= lim lim ( (-1)C(k’n)e
0 t--)O k=O
+ (--1)n-i.
Now the general manner in which we derived (0.10) suggests many other
numerical characters which may be associated either to singular varieties or to
a family of complex manifolds tending to a singular limit. The numbers which
are essentially new--i.e., are not expressible in terms of classical Milnor numbers-are
associated to an isolated singularity in codimension => 2, and arise
using either the st order tangential structure as for an isolated surface singularity
in 4, on the higher order osculating structure as for an isolated curve singularity
in (3. We don’t know if any among these has topological meaning, or if
any may be used to detect the impossibility of smoothing, or indeed any futher
non-trivial applications, but it seems pretty clear that additional work in the
area is possible, and so in 5d we conclude with a few examples analogous to
(0.9), general observations, and open questions.
Footnotes
1. L. Ness, Curvature of algebraic plane curves, I. Compositio Math., vol. 35 (1977), pp. 57-
63.
2. R. Langevin, Courbure et singularities complexes, to appear in Comm. Math. Helv.
3. Throughout this paper C will denote a suitable positive constant depending only on dimensions
and not on the manifold in question.
4. In fact, all computations are made in the setting of moving frames, so that the present work
may be considered as a continuation of our previous expository paper in this journal (vol. 41 (1974),
pp. 775-814). There we commented that the general theory of frames associated to higher order jets
was complicated by the presence of singularities arising from "inflectionary behaviour". In this
paper we will be eventually interested in singularities occurring already at the first order, so that its
general character may be said to be the application of frames to the metric study of complexanalytic
singularities.
5. J. Milnor, Singular points of complex hypersurfaces, Annals of Math Studies #61, Princeton
Univ. Press, Chapter 10.
6. S. S. Chern, On the kinematic formula in integral geometry, J. of Math. and Mech., vol 16
(1966), pp. 101-118.
Our discussion of integral geometry has been much influenced by unpublished course notes by
Chern from a class taught at Berkeley in the spring of 1965.
7. F. J. Flaherty, The Volume of a Tube in Complex Projective Space, Ill. Jour. Math., vol. 16
(1972), pp. 627-638.
1. Hermann Weyl’s formula for the volume of tubes
(a) Frames and derivation of the formula. We will discuss a formula of H.
Weyl