Page 190 - Curvature and Homology

172 V. COMPLEX MANIFOLDS

In particular,

Alejl A *** A eja A okl A **a A okr A Q)

Thus, for any p-form a, A(a A Sd) = Aa A 52 + (n - p)a. This result
will prove useful in the sequel.
consider the space C, of i complex variables with complex coordinates
zl, .-., zn and metric

Let a = ajl.. .,&.. ddl A A dda A dZk1 A .*- A dzkr and denote
by a, the operator which replaces each coefficient aj .. .jh.. .kr by the
coefficient of dal in daf , In a similar way we dehne the operator
&. The forms aim an&"::;';:: each of bidegree (p, r). Moreover, the
operators af and - 2$ are duals, that is, (a,@, /3) = - (a, E@). If we
put 8, = dzf, then

and, since 6' and 8" are dual to d' and d", respectively,

8 = - 2, i), 8'. = - a, @,),
i j
Consider, for example, the linear differential form a = afdzf + bfd9.
Then, since {dzi) and {a/azf) are dual bases