Page 186 - Curvature and Homology

168 V. COMPLEX MANIFOLDS

connection of a complex parallelisable manifold is holomorphic.) In
Chapter VI it is shown, if the manifold is simply connected, that this
condition is also sufficient. Hence, for a complex manifold the existence
of a metric with zero curvature is a somewhat weaker property than
parallelisability .

5.4. The operators L and A
Let M be a complex manifold of complex dimension n and denote
by A *C(M) the bundle of exterior differential polynomials with complex
values. From 5 5.1, a p-form a E A *C(M) may be represented as a sum

where a*,, is of degree q in the dzhnd of degree r in the conjugate
variables. The coefficients of a when expressed in terms of real coor-
dinates are complex-valued functions of class 00. Thus, there is a
canonically defined map

obtained from d by extending the latter to r\*C(IM) by linearity, that is,
if a = X + Flp where X and p are real forms, then

Clearly, d6 = &, that is d is a real operator. In the sequel, we shall
write d in place of d with no resulting confusion.
The exterior differential operator d maps a form a of bidegree (q, r)
into the sum of a form of bidegree (q + 1, r) and one of bidegree
(q, r + 1). For, if

The term of bidegree (q + 1, r) will be denoted by d'a and that of
bidegree (q, r + 1) by d"a. Symbolically we write