Page 220 - Curvature and Homology

202 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS

At the origin of this system of local complex coordinates glj. = atl.
Hence, from (5.3.19), a straightforward computation yields

and so from the covering of P, given in § 5.9, since
d' d" log To = d' d" log vj
for every index j = 1, ..a, n, the curvature tensor has this form every-
where. In other words, since there eiists a transitive Lie group of
holomorphic homeomorphisms preserving the metric, the curvature
tensor has the prescribed form everywhere.
Corollary. The holomorphic sectional curvature with respect to g of
complex projective space is positive.
An application of theorem 3.2.4 in conjunction with theorem 5.7.2
yields the betti numbers of a compact Kaehler manifold with the
Fubini metric (6.1.3) and, in particular, those of P,.

Theorem 6.1.4. The betti numbers b, of a compact Kaehler manifold M
of positive constant holomorphic curvature vanish if p is odd and are equal
to1 ifpiseeren:
btr = 1, bar+, =0, 0 br bn.

To see this, let fi be an effective harmonic p-form on M. Then,
is a harmonic p-form, and since

(cf. 5 5.4 for the definition of * for complex differential forms), it is
also effective. It follows that
.=g+P
is a real effective harmonic p-form. Now, put o = aAl. ..A9 d#l A A dad.
and compute F(a) (cf. formula (3.2.10)). In the first place, from
(6.1.2)