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CURVATURE AND COMPLEX SINGULARITIES 503
For example, suppose that we consider the basic Schubert cycle 2;’*; then
(- 1)’C t KdA
is the Gauss-Bonnet integrand (we use dA instead of dVt), and (5.11) is
(5.13) lim Iv Ct- KdA Ct- t-*0
Iv KdA + (-1)’* # (A,
If we assume that the origin is the only isolated singularity of V0 and set Vt[.]
ilzll _< then by the discussion in section 3(c)
which, when combined with (5.13) gives
KdA 0
(5.14) lim lim C te | KdA (-1)’*# (A, X’*).
0 0 Jvt[]
Now suppose that V0 c En + is a hypersurface with an isolated singularity
at the origin. If V0 is given by an analytic equation f(za, ", z’* / 1) 0, then
setting Vt {f(z) t} embeds V0 in a family {Vt} with Vt smooth for # 0. By
(4.10) the Schubert cycles 2;’* are in one-to-one correspondence with the hyperplanes
H through the origin, and by (4.12)
#(T(V), H) {number of times the tangent plane to V is parallel to H}.
We shall sketch the proof of the following result of Tessier: (6)
For sufficiently small and H generic,
[(n>}
__ (’*+ (5.15) # (y(Vt), H) # (y(Vo), H) + 1> {/x
where t (i) is the ith Milnor number of Vo.
Proof. We recall (z) that for e, t sufficiently small and n -> 2, Vt[e] has the
homotopy type of a wedge of n-spheres; the number of these is the top Milnor
number (n / a). The remaining Milnor numbers are defined by:/z ("- k / a) top
Milnor number of L N V0 where L is a generic (;N- k through the origin in N,
and where we agree to set/x (a) mult0(V0) and/x () 1. Choose a generic
linear coordinate system (z 1," ", z’*, u) such that the hyperplanes parallel to H
are given by u constant. The projection
u’V---,.
fibres Vt by varieties Vt,,, of dimension n 1, and the critical values//1, ", //K
correspond exactly to tangent hyperplanes Hu parallel to H. We may assume
that each Hu is simply tangent, and thus by Lefschetz theory (8) as u ux the
Vt,,, acquire an ordinary double point with there being a single vanishing cycle
5 H,_ l(Vt,0). The picture in the (real) (t, u) plane is something like