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while by Stokes’ theorem
CURVATURE AND COMPLEX SINGULARITIES 467
C te deg T
Ct--emulto(V),
7 )iT[r] 9Q .If[r] (’on
which tends to zero as r ---> 0, thus proving (3.17).
We note the general principle, already familiar from the use of polar coordinates
in elementary calculus, that blowing up frequently simplifies singular integrals.
In the next section we shall be considering varieties Vt acquiring an isolated
singularity at the origin and shall show that suitable curvature integrals
lim 0
2k
exist and have limits as e---> 0, and shall eventually give geometric interpretations
of these. For k n the above limit is
e2, vol V0[e], --
which then
tends to multo(V0) as e 0.
(c) Volume of tubes in the complex case. We shall now derive the formula
for the volume of the tube around a complex manifold M, C U. Proceeding as
in the real case, we let N[r] denote the ball of radius r around M embedded as
zero cross-section in its normal bundle, and by z(M) the image (counting multiplicities)
of the map
Points in the image are
N[r] --> ([N,
w z + Z t, e., litll-< r,
and from the structure equations (3.1) and (3.2)
dw (to. + tgto..)e. + (dt. + t(o.)e.
with repeated indices being summed. Setting
d to + t,to, toe t,h,(o
(recall that II h,oto (R) e, is the 2nd fundamental form ofM C U), the