Page 223 - Curvature and Homology

Since the even-dimensional betti numbers are each one and b,,, = b,,,
we conclude that

with all remaining b,,, zero. In particular,

b,,, = 0 for p # 0.
By employing the methods of theorem 3.2.7, it can be shown that a
4-dimensional &pinched compact Kaehler manifold is homologically P2
provided S is strictly greater than zero (st>dy positive curvature).
The reader is referred to V1.D for details. Hence, S2 x SZ considered
as a Kaehler manifold cannot be provided with a metric of strictly
positive curvature. In fact, it is still an open question as to whether
S2 x S2 can be given a Riemannian structure of strictly positive
curvature. For more recent results the reader is referred to [90] and [94l.
The n-sphere, complex projective n-space, quaternionic projective
n-space and the Cayley plane are the only known examples of compact,
simply connected manifolds which may be endowed with a Riemannian
structure of strictly positive curvature [I].

6.2. The effect of positive Ricci curvature

Since the Ricci curvature associated with the Fubini metric of P,, is
positive it is natural to ask if corollary 2 of the previous section can be
extended to any compact Kaehler-Einstein manifold with positive Ricci
curvature. An examination of the proof of theorem 6.1.4 reveals more,
however. For, if ,8 is a holomorphic form of degree p,

is a real p-form ; in fact, a is harmonic since /3 and j7 are harmonic. Hence,
since a is the sum of a form of bidegree (p, 0) and one of bidegree (0, p)
it follows from the symmetry properties of the curvature tensor that

Let M be a compact Kaehler manifold of positive definite Ricci
curvature. Then, by theorem 3i2.4, since a is harmonic, and F(u) is
positive definite, a must vanish.