Page 188 - Curvature and Homology

1 70 V. COMPLEX MANIFOLDS
so that, in general, (a, /3) is complex-valued. However, (a, a) 2 0,
eqaality holding, if and only if, a = 0. Two p-forms a and /3 are
said to be orthogonal if (a, /3) = 0. Evidently, if a and ,8 are pure forms
of different bidegrees they must be orthogonal.
The dual of a linear operator is defined as in's 2.9.
The operator * maps a form of bidegree (q, r) into a form of bidegree
(n - r, n - q). The dual of the exterior differential operator d is the
operator 6 which maps p-forms into (p - 1)-forms. We define operators
8' and 8" as follows:
6' =: -*&'* and 6" = -*&*
(cf. formula 2.8.7).
Clearly, then, 6' is of type (- 1,O) and 6" of type (0, - 1). Moreover,

For, 6 = -*d* = -*dt* -*dJ'*.
If M is compact or, one of a, /3 has a compact carrier,

(#a, P) = (a, S'P) and (Pa, 13) = (a, ti"@)
where a is a p-form and /3 a (p + 1)-form. For,

If or is of bidegree (q, r), /3 is of bidegree (q + 1, r); for, otherwise d'a
and /3 are orthogonal. In this way, it is evident that the desired relations
hold. Hence, 6' and 6" are the duals of d' and d", respectively.
Evidently,
6' 6' = 0, 6'' 6" = 0, 6' 6" + 6" 6' = 0.

In terms of the basis forms {BI) and {b} (i = 1, ..., n), the fundamental
form SZ is given by

We define the operator L on p-forms a of bidegree (q, r) as follows:

Hence, La is of bidegree (q + 1, r + l), that is, L is of type (1, 1). Fox
a p'-form /3
LaA*P=a AL*F=a A **-'L*P=(- 1)9'aA **L*/3.
We define an operator A of type (- 1, - 1) in terms of L as follows: