Page 166 - Curvature and Homology

1 48 V. COMPLEX MANIFOLDS

coordinates zl, ..-, zn. Then, the differentials dzl, -.., dzn constitute a
(complex) base for the differential forms of degree 1. It follows that a
differential form of degree p may be expressed in U as a linear
combination (with complex-valued coefficients of class w) of exterior
products of p-forms belonging to the sets {dzi} and {dZi}. A term
consisting of q of the {dzi} and r of the {dZi} with q + r = p is said
to be of bidegree (q, r). A differential form of bidegree (q, r) is a sum
of terms of bidegree (q, r). The notion of a form of bidegree (q, r) is
independent of the choice of local coordinates since in the overlap of
two coordinate neighborhoods the coordinates are related by holo-
morphic functions. A differential form on M is said to be of bidegree
(q, r) if it is of bidegree (q, r) in a neighborhood of each point.
It is now shown that a complex manifold is orientable. For, let zl, ..a, zn
be a system of local complex coordinates and set zk = sL + C y k .
Then, the xi and yJ together form a real system of local coordinates.
Since dzk A dZk = - 2 Gld2 A dyk,

It follows that the form

is real. That M is orientable is a consequence of the fact that 8 is defined
globally up to a positive factor. For, let wl, ..., wn be another system
of local complex coordinates. Then,

dwl A ... A dwn = Jdzl A ... A dzn
where
qwl, ..., wn)
J = det
qzl, ...,P)
Hence, dfil A ... A dGn = jdZ1 A ... A dZn from which

To define 8 globally we choose a locally finite covering and a partition
of unity subordinated to the covering.
We have seen that a complex manifold is by definition even dimensional
and have proved that it is orientable. These topological properties