Page 248 - Curvature and Homology
P. 248
230 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS
Thus,
where the SZk, are the forms Qk, pulled down to M. (The Qif. are defined
by the above relations.)
From 6.12.1 we deduce that
- - ln det (R,.)
-
(27r
N
- det (x Dka(;) Dp(;) dak A dzz).
-
__r
(27r m ) n 7-1
where, for simplicity, we have writen Qii, for its image in B (cf. 5 5.3).
Now, put
Then,
N
c1 = det (2 p(;)
(27r rn)" ,--1
where Q) is the matrix (&)) and 'Q) denotes its transpose.
The result follows after expressing Q) in terms of real analytic coordi-
nates (xi, yi) with ai = xi + d3 yi, since dai A dii = - 2 d*' A dyi.
Theorem 6.12.1. The Euler characteristic of a compact complex manifold
of complex dimmon n on which there exists N 2 n closed holomorphic
dz~erentials a$) dzg such that rank(a(i))= n satikfis the inequality
(- 1)" x(M) 2 0.
Moreover, x(M) vanishes, if: and only if, the nth Chern class vanishes [8].
6.13. The effect of sufficiently many holomorphic differentials
It was shown in 5 6.1 1 that the existence of sufficiently many inde-
pendent holomorphic differentials which are, at the same time, d'-closed
precludes the existence of holomorphic contravariant tensor fields of
