Page 193 - Curvature and Homology

5.6. Topology of a Kaehler manifold
The formulae (5.4.7) hold in a Kaehler manifold as one easily sees
by choosing a complex geodesic coordinate system (zi) at a point P.
Indeed, for C,, we may take g = 2 I=( dzi @ d9. Since the metric of a
Kaehler manifold has this form modulo terms of higher order, and
since only first order terms enter into the derivation of the formulae
(5.4.7) they must also hold in a Kaehler manifold.

Lemma 5.6.1. In a Kaehler manifold

Ad' - d'A = - l/---Zi
and (5.6.1)
Ad" - d"A = 6 1 6'.
These formulae are of fundamental importance in determining the
basic topological properties of compact Kaehler manifolds.

Lemma 5.6.2. In a Kaehler manifold the operators A and 6 commute.
Hence, by comparing types A commutes with 8' and 6".
Clearly, the operators L and d commute. Hence,
*d**-l L* = *L**-1 d*,
that is
SA = 116.
Several interesting consequences may be derived from lemmas 5.6.1
and 5.6.2 for a complex manifold with a Kaehler metric. To begin with
we have

Lemma 5.6.3. In a Kaehler manifold

d' 6" + 6" d' = 0 and d" 6' + 6' d = 0.
The proof is immediate from lemma 5.6.1.

Lemma 5.6.4. In a Kaehler manifold

For, from lemma 5.6.1 the expression
- <l(d'~d"- d"Adl + dUd'A - Ad'd) is equal to 8' 6" + 6" d"
from the first relation and to d' 6' + 6' d' from the second.