Page 252 - Curvature and Homology

234 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS
Lemma 6.14.1 [47]. The real closed 2-form

on M is a representative of the characteristic class c(B). Conversely, if y
is a real closed form of bidegree (1,I) on M belonging to the characteristic
class c(B), there exists a system of positive functions a, of class oo such
that for each pair at, /3

and
47 a2 log a,
y=-- azi a59 dzi A dzj.
27T
The 2-form y is said to be positive (y > 0) if the corresponding
hermitian quadratic form is positive definite at each point of M. Let

be a differential form of bidegree (p, q) with coefficients in B and denote
by P*q(y, v) the quadratic form (corresponding to F(at) in 5 3.2-the
operator A being given by A = 2(d'6' + 6'df)),

-orks.. .kpp ,:. . .p68kr.-kT ii--i:
where yiej, = gM' ykj..
We now state the vanishing theorems:
Theorem 6.14.1. If the characteristic class c(B) contains a real closed form

with the property that the guadratic form P*q(y, v) is positive definite
at each point of M, then
HQ(M, A '(B)) = (01, q = 1, -*, n.

Theorem 6.14.2. If the form y > 0,
HQ(M, A~(B)) = {o}, q = 1, -, n.