Page 215 - Curvature and Homology

CHAPTER VI

CURVATURE AND HOMOLOGY
OF KAEHLER MANIFOLDS

It is a classical theorem that compact Riemann surfaces belong to
one of three classes (cf. example 1, § 5.9). However, for several complex
variables the situation is not quite so simple. In any case, there is the
following generalization, namely, if M is a compact Kaehler manifold
of constant holomorphic curvature k (cf. 5 6.1), its universal covering
space is either complex projective space P,(k > O), the interior of a
unit sphere B,(k < 0), or the space Cn of n complex variables (k = 0).
These spaces are of interest in algebraic geometry; indeed, they provide
a source of examples of algebraic varieties. In analogy with the real case
(cf. 5 3.1) a (compact) Kaehler manifold of constant holomorphic
curvature is called elliptic, if k > 0, hyperbolic, if k < 0 and parabolic
if k = 0. By an appIication of the results of Chapter V it is shown that an
elliptic space is homologically equivalent to complex projective space.
It is, in fact known, in this case, that M is actually Pn itself. If the
manifold M is parabolic it can be represented as the quotient space
C,/D where D is a discrete group of motions in Cn, namely, the
fundamental group. The group r in example 5, 5 5.1 is a normal sub-
group of D of finite index with 2n independent generators. The complex
torus Tn = Cn/r is then a covering space of M.
On the 1-dimensional (complex) torus TI there is essentially only one
holomorphic differential, namely, dz in contrast with the Riemann
sphere on which none exist (cf. 5 5.6). In higher dimensions there is the
analogous situation, that is, on Tn there are n independent holomorphic
pfaffian forms whereas in the elliptic case there are no holomorphic
1-forms. More generally, on a compact Kaehler manifold of positive
definite Ricci curvature, there do not exist holomorphic p-forms
(0 < p 5 n) [58].
The reader is referred to 5 5.9, example 3 for a description of the
complex torus. Now, the torus has 'zero curvature' and this fact is decisive