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As a consequence of (5.15),
CURVATURE AND COMPLEX SINGULARITIES 505
lim # (T(Vt), H)dH t" # (y(V0), H)dH + tz ("+ 1) + /.(.),
which when combined with (5.12) and (5.14) gives Langevin’s theorem (0.9)
(5.17) lim lim C te [ KdA (-1)"{/ TM + 1) + /z(,)}.
e--) O t--* v
(c) Extension to higher codimension and isolation of the top Milnor number.
Even though our main formula (5.11) is fairly general in scope, it is clearly of
the same character as the special case (5.17). The extension to higher codimension
is perhaps more novel, relying as it does on the formula (4.28) instead
of the simpler Crofton formula (4.9).
As above we assume given {Vt} where Vt is smooth for # 0 while V0 has an
isolated singularity at the origin. For a generic linear space L G(N k, N),
the intersections Vt D L will be transverse and therefore smooth for # 0
while V0 D L C L will have an isolated singularity at the origin. We may define
the corresponding Plficker defect AL, which is then an analytic subvariety of
G(n k, L) G(n k, N k). By (5.8) and (5.15)
(5.18) lim cn-k(12vt n L)= c-k(fv0 n )
Jvt f- L JVo f-I L
+ {p("- + ) + t("- )}.
For : 0 the Crofton’s formula (III) (4.28) gives
(5.19) I ( jt c (fvt n )) dL C fv cn (fv) / t"
To evaluate the right hand side we use the
LEMMA: For O and tO a closed (n k, n k) form on Vt,
(520). fv b / fv
Proof. Referring to the notation in the proof of (3.15), we set
o- / d e log Ilzll z/ to- 1.
Then do- tO/ to, while on the boundary O V[]
d log
(cf. the computation following (3.16)). By Stokes’ theorem, which may be applied
since V is smooth for 0,
t[e]
Vt[d