Page 239 - Curvature and Homology

group of those homeomorphisms o of I@ with itself such that n - o = n
for every element a E D. Then,

where o* is the induced dual map on A*c(.@). Hence,

that is the & are invariant under D. It follows that a is a left translation
of m, and so D may be considered as a discrete subgroup of the complex
Lie group f@. With this identification of D, M is holomorphically
isomorphic with MID. Thus,
Theorem 6.9.1. A compact complex parallelisable manifold is holo-
morphically isomorphic with a complex quotient space of u complex Lie
group modulo a discrete subgroup [69].
I
Corollary. A compact complex parallelisable manifold is Kaehbian, if
and only if, it is a complex multi-torus.
A complex torus is compact, Kaehlerian, and complex parallelisable
(cf. example 3, 5 5.9). Conversely, if M = G/D is Kaehlerian, the left '
invariant pfaffian forms on the complex Lie group G must be closed.
It follows that G is abelian. Therefore, M is a complex torus.
Theorem 6.9.1 may be strengthened by virtue of theorem 6.7.1. For,
zero curvature alone implies that the TI> satisfy the equations of
Maurer-Cartan. It follows that the 8t are the left invariant pfaffian forms
of a local Lie group.

Theorem 6.9.2. A compact hermitian manifold of zero curvature is
holomorphically isomorphic with a complex quotient space of a complex Lie
group modulo a discrete subgroup.
Corollary. A compact hermitian manqold M of zero curvature cannot be
simply connected.
For, otherwise the left invariant pfaffian forms on M are closed.
Thus, M is an abelian Lie group, and hence is a complex torus. This,
of course is impossible. -

We have seen that d" is a differential operator on the graded module
A *"(M) (cf. 5 5.4) where M is a complex manifold. In this way, since
dU2 = 0, it is possible to define a cohomology theory analogous to the