Page 250 - Curvature and Homology

Theorem 6.1 3.1. Let ar, r = 1, -.-, N be N > n holomorphic dzyerentials
on the compact complex manifold M with the property : For any system
of constants cr, r = 1, ---, N(not all zero) the rank of the matrix (a(:), cr), r =
1, -.-, N; i = 1, -.., n has its maximum value n + 1 at some point. Then,
there do not exist (non-trivial) holomorphic contravariant tensor Jields of
any order on M. In particular, there are no holomorphic vector fields
on M [9].
This result is generalized in Chapter VII. In particular, it is shown
that if b,,,(M) = 2, M cannot admit a transitive Lie group of holo-
morphic homeomorphisms.

6.14. The vanishing theorems of Kodaira

A complex line bundle B over a Kaehler manifold M (of complex
dimension a) is an analytic fibre bundle over M with fibre C-the complex
numbers and structural group the multiplicative group of complex
numbers acting on C. Let Aq(B) be the 'sheaf' (cf. 5 A.2 with r =
Ag(B)) over M of germs of holorhorphic q-forms with coefficients in B
(see below). Denote by Hp(M, Ag(B)) the pth cohomology group of M
with coefficients in Aq(B) (in the sense of 5 A.2). It is known that these
groups are finite dimensional [47]. It is important in the applications of
sheaf theory to complex manifolds to determine when the cohomology
groups vanish. By employing the methods of 5 3.2, Kodaira [47] was
able to obtain sufficient conditions for the vanishing of the groups
Hp(M, Aq(B)). It is the purpose of this section to state these conditions
in a form which indicates the connection with the results of 5 3.2. The
details have been omitted for technical reasons-the reader being
referred to the appropriate references, principally [97].
In terms of a sufficiently fine locally finite covering 4"1 = {Ua} of M
(cf. Appendix A), the bundle B is determined by the system Cf,B)
of holomorphic functions fa,, (the transition functions) defined in Ua n Ue
for each a, /3. In Ua n Ue n U,,, they satisfy fae fey fya = 1. Setting
-- 1 faS la, it is seen that the functions {aaJ define a principal fibre
'a8 -
bundle over M (cf. I. J) with structural group the multiplicative group
of positive real numbers. This bundle is topologically a product. Hence,
we can find a system of positive real functions {aa} of class m defined in
(UJ such that, for each pair a, /3