Page 217 - Curvature and Homology

6.1. HOLOMORPHIC CURVATURE 199
are 'sufficiently many' holomorphic pfaffian forms, then (- l), x(M) 2 0
where X(M) is the Euler characteristic. An example is provided by T,
for which it is clear that x(T,) = 0 [8].
Denote by the pair (M, g) a Kaehler manifold with metricg and under-
lying complex manifold M. Consider the Kaehler manifolds (M, g)
and (M, g'). If the connections o and o' canonically defined by g and g',
respectively, are projectively related, a certain tensor w (the complex
analogue of the Weyl projective curvature tensor) is an invariant of
these connections. Its vanishing is of interest. For, if w = 0, the manifold
(M, g) (or (M, g')) has constant holomorphic curvature. Conversely, for a
manifold of constant holomorphic curvature, w = 0. In this way, constant
holomorphic curvature is seen to be the complex analogue of constant
curvature in a Riemannian manifold [33]. (A Kaehler manifold of
constant curvature is of zero curvature). The homological structure of
elliptic space is, as previously mentioned, identical with that of P,.
However, the betti numbers of P, are retained even for deviations from
projective flatness [7].
An important application of the results of Chapter I11 is sketched
in 5 6.14 where the so-called vanishing theorems of Kodaira are obtained.
These theorems are of interest in the applications of sheaf theory
to complex manifolds since it is important to know when certain
cohomology groups vanish.

6.1. Holomorphic curvature

Let M be a Kaehler manifold of constant curvature K whose complex
dimension is n. Then, from (1.10.4) the curvature tensor is given by

(The same systems of indices as in Chapter V are maintained throughout.)
In terms of local complex coordinates these equations take* the form

Substitution of this last set of equations into (5.3.39) gives