Page 194 - Curvature and Homology

1 76 V. COMPLEX MANIFOLDS

Lemma 5.6.5. In a Kaehler manifold the Laplace-Beltrami operator
A = d 6 + 6 d has the expressions
A = 2(d' 8' + 8' d') = 2(d" 6" + 6" d). (5.6.3)
For,
A = d8 + Sd
= (d' + d") (8' + 8") + (8' + Sf') (d' + d")
= (d' 8' + 8' d') + (d" 8" + 8" d")
by lemma 5.6.3. Applying lemma 5.6.4, the result follows.
A complex p-form a is called harmonic if Aa vanishes.
Since a p-form may be written as a sum of forms of bidegree (q, r)
with q + r = p we have:

Lemma 5.6.6. A p-fanr is har11~0l[tic, if and only if its various tmns of
bidegree (q, r) with q + r = p are 'ehamumic.
This follows from the fact that A is an operator of type (0,O). Indeed,
d' is of type (1,O) and 8' of type (- 1,O). Moreover, a p-form is zero, if
and only if its various terms of bidegree (q, r) vanish.

Lemma 5.6.7. In a Kaehler manz~old A commutes with L and A. Hence,
if a is a harmonic form so are La and Aa.
This follows easily from lemmas 5.6.1 and 5.6.2 since 8' 8" + 8" 8'
= 0 and *A =A*.

Lemma 5.6.8. In a Kaehler mantfold the fm Q" = Sa A A S2
(p times) fw every integer p 5 n are harmonic of &pee 2p.
The proof is by induction. In the first place, ASZ = 0 since the manifold
is Kaehlerian. For, by lemma 5.6.1,tjtSZ = 8"Q = 0 since d'S2 = d"S2 =
0 and WE = n. Now,

Lemma 5.6.9. The cohomology groups H@(M, C) of a compact Kaehler
manifold M with complex coeficients C are dtferent from zero for
p = 0, 1, -, n.
Indeed, by the results of Chapter 11, Ha(M, C) is isomorphic with
the space of the (complex) harmonic forms of degree q on M. Since
the constant functions are harmonic of degree 0, the lemma is proved
for p = 0. The proof is completed by applying the previous lemma
and showing that LP # 0 for p < n. In fact, we need only show that
LP # 0, and this is so, since SZ" defines an orientation of M (cf. 5 5.1).