Page 230 - Curvature and Homology

2 12 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS

then, since t + t is self adjoint, f is a real-valued function, and since the
opetator A is real, Af is real-valued. Moreover,

f
f
- *Af = figABDBDA = g(j*Dj.Di
- + G(t) (6.5.3)
- gilr; ...gi9~g~1'~..g~~d~gk1*~ktil~~~i~jII,,jQ~1tr1111p~8111,~Q
where

Expanding G(t) by (6.5.2) gives

Since the first term on the right in 6.5.3 is non-negative we may conclude
that Af 5 0, provided we assume that the function G is non-negative.
Hence, as a consequence of proposition 6.5.1

Theorem 6.5.1. Let t be a holomorphic tensorjkld of type (t @. Then, a
necessary and suflcient condition that the (self adjoint) tensorfield t + f
on a compact Kaehler manifold be parallel is given by the inequality

On the other hand, if G(t) is positive somewhere, t must vanish, that is
there exists no holomorphic tensor field of the prescribed type [Ill.
An analysis of the expression (6.5.4) for the function G yields without
difficulty

Corollary 1. Let t be a holomorphic tensor jkld of type (t g) on the
compact Kaehler manifold M. Then, if

t is a parallel tensor field. If strict inequality holds at least at one point of M,
t must vanish.
If M is an Einstein space,