Page 232 - Curvature and Homology

2 14 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS

terms of these pfaffian forms, the fundamental form $2 has the expression

If we put (cf. V, B. 1)

then, by (3.5.3) and (3.5.4)

Hence, since the Lie algebra of holomorphic vector fields is abelian,
the Oi are d'-closed for all i = 1, n. (Referring to the proof of theorem
6.7.3, we see that they are also d"-closed.) This being the case for the
conjugate forms as well

that is, M with the metric 2 ZOr @ 8' is a Kaehler manifold. Moreover,
the fact that the Bi are closed allows us to conclude that M is locally
flat. To see this, consider the second of the equations of structure
(5.3.33):
sij = deij - ek, A ei,.

Taking the exterior product of these equations by Oj (actually .rr*Oj,
cf. 5 5.3) and summing with respect to the index j, we obtain

Indeed, from the first of the equations of structure (5.3.32), Oj A Okj
vanishes since the Oi are closed. Moreover,

ej A dekj = - d(8' A ekj) = 0.

If we pull the forms eij down to M and apply the equations (5.3.34),
we obtain
Riikl* dzk r\ dZ1 A dzj = 0,

and so, since the curvature tensor is symmetric in j and k, it must vanish.
In $6.9, it is shown that if M is compact, it is a complex torus.