Page 256 - Curvature and Homology

238 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS
3. The manifold M is said to be A-holomorphically pinched if, for any holo-
morphic section there exists a positive real number Kl (depending on g)

The metric g may be normalized so that Kl = 1, in which case,

A &pinched Raehler manifold is S(88 + l)/(l - 6) -holomorphically
pinched.
To see this, apply the inequality
I Rijkl I 5 f [(PS)"' + (QR)"'] (1)

i, j,k,l = 1, -, 2n where
valid for any orthonormal set of vectors {Xi,~j,~k,~,},

This inequality is proved in a manner analogous to that of (3.2.21); indeed, set
L(a,i;b,k;c, j;d,l) = G(a,i;b,k;c, j;d,l) + G(a,i;b,l;c, j; - d,k)
and show that
L = Pa2$ + Qa2dq Rb2c2 + SbZd2 + 6Rijkl abcd.

Put Xs* = JXi(i = 1, ..., n) (cf. (5.2.6)) and apply (1) with j = i* and 1 = k*.
Hence, from 1 .(b) - (d)

from which
[(Kip - 6) (Kkp - 6)]1/2 2 Kfk + Kip + 4.

Since Ki, 2 6, Kik, 2 6 and Kk,, $ 1, we conclude that

(Note that a manifold of constant holomorphic curvature is 4 - pinched.)
4. Prove that if M is A-holomorphically pinched, then M is 3(7X-5)/8(4-A)-
pinched.
In the first place, for any orthonormal vectors X and Y, g(aX + by, ax + by)
= az + b2. Applying l.(b) and (c) as well as (1.1. l(d)),
(a2 + b2)2K(aX + b Y, J(aX + b Y)) = dK(X, ]X) + b4K(Y, JY)
+ 2a2b2[K(X, Y) - 3g(R(Y, JX)Y, JX)]
+ 4a&(R(Y, JY)X, JY) + 4aJbg(R(X, ]x)x, IY).