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504 PHILLIP A. GRIFFITHS where D is the discriminant curve of points (t, u) such that Vt,u is singular. The picture in the complex u-plane is where the ux(t) are the intersections of D with the vertical line constant. It follows that the Euler characteristics are related by (5.16) x(Vt) x(Vt,o) + (-1) On the other hand, Vt Vt[e] and Vt,o Vt,0[e] may be assumed homotopic for e, sufficiently small while Together with (5.16) this implies that which proves (5.15). x(Vt[e]) ("+ 1) + (-1)"/ x(Vt,o[e]) + (-I) n-
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
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Vol. 45, No. 3 DUKE MATHEMATICAL JO
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CURVATURE AND COMPLEX SINGULARITIES
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we deduce that CURVATURE AND COMPLE
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so that CURVATURE AND COMPLEX SINGU
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