View PDF - Project Euclid

we conclude that
CURVATURE AND COMPLEX SINGULARITIES 471
c(flv,)A dpn-= f C(E)Adpn- J*
converges as asserted.
We remark that, upon approaching a singular point z V, as noted above
the Gauss mapping may or may not extend according as to whether or not the
tangent planes T,(V*) have a unique limiting position for all arcs z(t)
--
in V*
which tend to z. Thus when dim V 1 it always extends, and in this case it may
or may not happen that the Gaussian curvature K(z(t)) -o as z(t) z. In
fact the Gaussian curvature remains finite if, and only if, z is an ordinary singularity
of V.
Now we assume that V is smooth except possibly at the origin. Then we will
prove that the function
(3,20) Ik(U, r)
(-1) I c(tv) A n-
r2n- 2k
is an increasing function of r(1). For this we use the notations of section (b)
above. By (.6) the differential form (-l) c(f,) is C, closed, and non-negative
on V* V {0}. Since o d d e log iiz[i is the pullback of the Kfihler form
on IP N- 1,
(3.21) (-1) k ck(l)v) A o n- >- O.
[p,r]
On the other hand, setting
[r]
de I1 11 A
we have from d o and Stokes’ theorem that
(- 1) I ck(t]v) A on (- 1) c(lv) A
v [p,r] V[r]
But on the sphere Ilzll
and therefore
’Ok
(-) c(a) A
ov[p]
Ck(’N) A "Ok t2n- 2k
v[t] v[t]
t2n 2k c(t2v) A (b n- .
It]