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470 PHILLIP A. GRIFFITHS
We should like to make a few observations concerning this formula. The first
is that an analogous result holds for a complex manifold Mn C IP u (8), more
precisely, the formula is
(3.19) vol rdM)
where
/=0
I 2(m C(k, n, m)r + ) P(ft, 6)/ 6"-
Pk(12M, 6) C(1, k) C,(M) /k dp- t.
/=0
In case M is compactmi.e., is a projective algebraic varietymthe integrals in
(3.19) depend only on the tangential Chern classes cq(M) Hq(M) and hyperplane
class H(M). So we conclude that, just as the Wirtinger theorem
(3.14) implies that the volume of M is equal to (a constant times) its degree
", the formula (3.19) implies that the volume of the tube is again of a topo-
M
logical character--as noted above, this is in strong contrast to the real case.
The second remark concerns the volume of the tube near a singularity of an
analytic variety V, C ([N. Setting B[r] {[Izl] 0 and denote by V that part of the variety in
the unit ball. Then we claim that the integrals 9
V
CK(-V*) / +n- tc
converge, and consequently the volume of the tube around a singular variety is
finite. To establish this, let
FCV G(n,N)
--
be the closure of the graph of the Gauss map
7 V* G(n, N).
It is easy to see that F is an analytic variety of pure dimension n, and that the
projection
is an isomorphism on that part F* lying over V*. In general there will be blowing
down over the singular set of V, since for z0 V the fibre 7r-l(z0) C G(n, N)
is the limiting position of tangent planes Tzt)(V*) along all analytic arcs {z(t)} C
V* with lim z(t) zo. Now the forms b"- / c(O) are well-defined on
t--0
U X G(n, N), and hence by (3.12) are integrable over the smooth points of F.
But then since