Page 253 - Curvature and Homology

The proof of theorem 6.14.1 is an immediate consequence of the fact
that Hq(M, AP(B)) g Hppq(B). For, by lemma 6.14.1, we may choose
the system of functions {aa} satisfying a, = I fd 12ap in such a way that
(471272) (a2 log aa/ azi Zj) dzi A d9 = y (cf. VI. H. 2). Then, by
the argument given below Fp.q(y, 4a) = 0, q = I, a-, n holds for any
form 4 = {+,I E Hppq(B). The result now follows since FP*q(y, 4J > 0
unless 4, vanishes.
Let - B denote the complex line bundle defined by the system
{f;;}. Then, the map 4 + 4' defined by
' 1
+a=- *Ba
a a
maps Hp,q(B) isomorphically onto Hn-Ppn-q(- B). Hence,
HQ(M, A "(B)) -- Hn-q(M, A n-p(- R)).

Corollary 6.1 4.1. Under the hypothesis of theorem 6.14.1
Hn-q(M, An-P(- B)) = (01, q = 1, *.-, n.
Corollary 6.14.2. If the form y > 0,
Hn-q(M, AO(- B)) = {0), q = 1, ..-, n.
By the canonical bundle,K over M is meant the complex line bundle
defined by the system of Jacobian matrices ka8 = a(z;, -, z~)/a(z~, --p:),
where the (2:) are complex coordinates in Ua. It can be shown that the
characteristic class c(- K) of - K is equal to the first Chern class of M.
The characteristic class c(B) is said to be positive definite if it can be
represented by a positive real closed form of bidegree (1,l). We are now
in a position to state the following generalization of theorem 6.2.1.
Theorem 6.14.3. There are no (non-trival) holomorphic p-forms (0 < p
5 n) on a compact Kaehler manifold with positive definite first Chern class.
This is almost an immediate consequence of theorem 6.14.2 (cf. 1471).
It is an open question whether there exists a compact Kaehler manifold
I
with positive definite first Chern class whose Ricci curvature is not
positive definite.
Proof of Theorem 6.14.1. Since M is compact, the requirement that
E HPpq(B) is given by the equations d"4, = 6"4, = 0 for each a.
In the local complex coordinates (zi), 4, has the expression