Page 200 - Curvature and Homology

182 V. COMPLEX MANIFOLDS

The difference bp - bp-, may be measured in terms of the number ep
of effective harmonic forms of degree p, p 5 n + 1 and is given by
the following

Theorem 5.7.2. On a compact Kaehler mani$old
e, = b, - b,-,
for p 5 n+ 1.
To see this, denote by A& the linear subspace of of effective
harmonic p-forms. Then, by corollary 5.7.3
A:, = A= LA&-;-' @ ... @ LrAg2r (5.7.4)

where r = Cp/2], and

where r = [p/2] + 1.
Applying the operator L to the relation (5.7.4) we obtain

Combining (5.7.5) and (5.7.6)
A&+2 = /I,+, @LA)
H H'
Since L is an isomorphism from A'(Tc*) into AV+~(TC*) j - 1) and
n
(p
since A commutes with L, dim LA& = dim A&. Hence,
dim ARC2 = dim i\gP + dim A&

that is bp+, = eP+, + bp,p 5 n - 1 or b, - bp-, = ep forp n + 1.

5.8. Holomorphic maps. l nduced structures

Let M and M' be complex~manifolds. A differentiable map f : M -+ M'
is said to be a holomorphic map if the induced dual map f*: A *C(M') +
A*'(M) sends forms of bidegree (1,O) into forms of bidegree (1,O).
Under the circumstances, f* preserves types, that is, it maps forms of
bidegree (q, r) on M' into forms of bidegree (q, r) on M. For, since f*
is a ring homomorphism we need only examine its effect on the decom-
posable forms (cf. 1 S).