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5.7. EFFECTIVE FORMS ON AN HERMITIAN MANIFOLD 1 79
where the operators H and G are the complex extensions of the cor-
responding real operators. Moreover, since the Green's operator G
commutes with d and S it commutes with d', d", Sf3 St' as one sees by
comparing types.
Since A commutes with d, it also commutes with d' and d" as one
sees by comparing types. This result is very important since it relates
harmonic forms with the cohomology theories arising from d' and d".

5.7. Effective forms on an hermitian manifold

There is a special class of forms defined as the zeros of the operator A
on the (1inear)space of harmonic forms. They are called effective harmonic
forms and the dimension of the space determined by them is a topological
invariant. More precisely, the number e, of linearly independent effective
harmonic forms of degree p on a compact Kaehler manifold M is
equal to the differenct bp - bp-, for p n + 1 where dim M = 2n. This
important result hinges on a relation measuring the defect of the operator
LkA from ALk where Lkor = or A @. The fact that these operators
do not commute is crucial for the determination of the invariants $.

Lemma 5.7.1. For any p-form or on an hwmitian manifold M

It was shown in § 5.4 that

Hence, proceeding by induction on the integer k

This completes the proof.
In the remainder of this section a subscript on a given form will
indicate its degree; thus deg cq, = p.
A form a is said to be effective if it is a zero of the operator A, that is,
if nor = 0. Since A annihilates ~(Tc*) for p = 0,l the elements of
these spaces are effective.