Page 187 - Curvature and Homology

5.4. THE OPERATORS L AND A 169

and say that d' is of type (1,O) and d" of type (0,l). By linearity, we
extend d' and d" to all forms. (An operator on A *C(M) is said to be
of type (a, b) if it maps a form of bidegree (q, r) into a form of bidegree
(q + a, r + b)). Both d' and d" are complex operators, that is if a is
real, d'a and d"a are complex.
Since
0 = dd = d'd' + (d'd" + dud') + dud"
it follows, by compaf ng types, that

and
d'd" + dud' = 0.

We remark that the operators d' and d" define cohomology theories in
the same manner as d gives rise to the de Rham cohomology (cf. 6.10).
Iff is a holomorphic function on M, dl'fvanishes. A holomorphic form
a of degree p is a form of bidegree (p, 0) whose coefficients relative to
local complex coordinates are holomorphic functions. This may be
expressed simply, by the condition, d"a = 0. It follows that a closed
form of bidegree (p, 0) is a holomorphic form.
At this point it is convenient to make a slight change in notation
writing Of in place of 8t.
Let (O,, ..-, be a base for the forms of bidegree (1,O) on M. Then,
8,)
the conjugate forms 8,, ..., 8, comprise a base for the forms of bidegree
(0, 1). Suppose M has a metric g (locally) expressible in the form

The operator * may then be defined in terms of the given metric.
Our procedure is actually the following: As originally defined * was
applied to real forms and played an essential role in the definition of the
global scalar product on a compact manifold or, on an arbitrary
Riemannian manifold when one of the forms has a compact carrier.
In order that the properties of the global scalar product be maintained
we extend * to complex differential forms by linearity, that is

Hence, if M is compact (or, one of a, has a compact carrier) we define
the global scalar product