Page 195 - Curvature and Homology

Theorem 5.6.1. A holomorphic form on a Kaehk manifold is harmonic.
For, if a is a holomorphic form, it is of bidegree (p, 0); moreover
d"ar vanishes. Now, since 8" is an operator of type (0, - I), S"a is a
form of bidegree (P, - l), that is S"a = 0. It follows that

Corollary. A holomorphic form on a compact Kaehk manifold is closed.
Conversely, a harmonic form of bidegree (p, 0) on a compact hermitian
manifold is holomorphic. For, a harmonic form is closed and a closed
form of bidegree (p, 0) is holomorphic.
The term of bidegree (P, 0) of a harmonic p-form ar is holomorphic.
Similarly, the conjugate of the term of bidegree (0, p) is holomorphic.
For, let

the subscripts indicating the bidegree. Then, since ar is harmonic and
the manifold is compact

Since the terms on the left side of this equation are of different bidegrees
they must vanish individually. In particular,

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Similarly, dl%, = 0 implies d'ao,, = d''a,,, = 0.
Let Ag be the linear space of complex harmonic forms of degree p.
Then, by lemma 5.6.6, A$ is the direct sum of the subspaces Ag of
the harmonic forms of bidegree (q, r) with q + r = P. The pth b,etti
number bJM) of the Kaehler manifold M is equal to the sum

where b,,, is the complex dimension of Aff. Now, if a E A%', its
conjugate 6 E A>,, and conversely. For,