Page 221 - Curvature and Homology

Next, we derive an explicit fornula for Aar in local complex
coordinates (zi). From (5.4.2) and (5.3.12)

Hence, since the interior product operator is an anti-derivation

Returning to equation (6.1 S), we conclude that
RABCDaABA."'APaCDAsI .AP = - 2kaij*As"'AVaij* .Ap .
I
Combining (6.1.4) and (6.1 S), the quadratic form
p
F(a) = (n + 1) koijAs-A~aijA,..Ap + (n - + 2) kaij*As-.A~a. *A,...A,
rj
>0, OcpSn,
that is, there are no non-trivial real effective harmonic p-forms for
p 5 n. Hence, by theorem 5.7.2

6,-, = b,, p 5 n + 1.
Now, by theorem 3.2.1, since the Ricci curvature is positive definite
(by virtue of the fact that k is positive), b, vanishes. Thus
b2,41 =0, 2r Sn.
On the other hand, since M is connected, b, = 1, and so

The desired conclusion then follows by Poincare duality.