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464 PHILLIP A. GRIFFITHS
and, for any affine n-plane L in N
(3.13)
n!
Euclidean measure on
(to see this choose orthonormal coordinates so that L is a translate of ([;’9. Two
important consequences of (3.13) are: (i) For any complex manifold M
(3.14)
n! b ’ vol (M)
is the Euclidean volume of M (Wirtinger theorem), and (ii) for any a A"(
there is an estimate
in the sense that for all complex n-planes L
where the function on the Grassmannian is bounded. The fact that (3.12)
defines a current is then equivalent to the finiteness of the volume
n Cl B[Z,]
<
of analytic varieties in the e-ball B[z, e] around singular points z V,. Similarly,
Stokes’ theorem follows from the usual version for manifolds together with
the fact that the (2n 1)-dimensional area of the boundary 0 T,(V,) of the e-tube
around the singular points tends to zero as e 0.
Later on we shall be examining more delicate integrals / where is not
jv
the restriction of a form in U (such as a curvature integral), or may be the
restriction of a form but one having singularities on V. To obtain some feeling
for these We shall examine one of the latter types here. With the notations
d c
we shall prove the formula (
-1
4
w d d c log ilzll
{z e v: ilzll r}
o, d r]- o], o r,
(3.15) r Oe
Geometrically, if we consider the residual map
[rl [O] [O,rl
CN {0} IP N
m.