Page 226 - Curvature and Homology

Theorem 6.3.1. In a compact Kaehler manifold M of complex dimension n
with positive de3nite Ricci curvature, if

fw allp = 1, -..,n where
(96 s>
h, = inf-
6 <&5>
the greatest lower bound being taken over all (non-trivial) forms of degree I,
M is homologically equivalent with P, [a.
The idea of the proof, as in theorem 6.1.4, is to show that under the
circumstances there can be no non-trivial effective harmonic p-forms
on M for p 5 n. Once this is accomplished the result follows by PoincarC
duality.
Let a = aA ... ,,dzAl A --- A dsd9 be a real effective harmonic
p-form on M. ?hen, from (3.2. lo), (6.3.3), and (6.3.6),

Since A,, > 0 the desired conclusion follows.
Corollary. Under the conditions of the theorem, if

M is homologically equivalent with complex projective n-space.

6.4. Kaehler-Einstein spaces
In a manifold of constant holomorphic curvature k, the general section-
al curvature K is dependent, in a certain sense, upon the value of the
constant k. In fact, if k > 0 (< 0), so is K; moreover, the ratio of the