Page 165 - Curvature and Homology

5.1 COMPLEX MANIFOLDS 147

5.1. Complex manifolds
A complex analytic or, simply, a complex manifold of complex dimension
n is a 2n-dimensional topological manifold endowed with a complex
analytic structure. This concept may be defined in the same way as the
concept of a differentiable structure (cf. $ 1.1)-the notion of a holo-
morphic function replacing that of a differentiable function. Indeed,
a separable Hausdorff space M is said to have a complex analytic structure,
or, simply, a complex structure if it possesses the properties:
(i) Each point of M has an open neighborhood homeomorphic
with an open subset in Cn, the (number) space of n complex variables;
that is, there is a finite or countable open covering {U,), and for each
a, a homeomorphism u, : U, -+ C,,
;
(ii) For any two open sets U, and Up with non-empty intersection
the map up;l : ua(Ua n Up) --+ Cn is defined by holomorphic functions
of the complex coordinates with non-vanishing Jacobian.
The n complex functions defining u, are called local complex coordinates
in U,. The concept of a holomorphic function on M or on an open subset
of M is defined in the obvious way (cf. V.A.). Every open subset of M
has a complex structure, namely, the complex structure induced by
that of M (cf. $ 5.8).
A complex manifold possesses an unhlying real analytic structure.
Indeed, corresponding to local complex coordinates zl, .-., zn we have
real coordinates xl, as-, xn, yl, -.. , yn where

moreover, in the overlap of two coordinate neighborhoods the real
coordinates are relateai by analytic functions with non-vanishing
Jacobian (cf. V.A.).
Any real analytic function may be expressed as a formal power
series in zl, , zn, i?, , Zn by putting

where Zk denotes the complex -conjugate of zk. Consequently, whenever
real analytic coordinates are required we may employ the coordinates
.$, ... , zfi, 51, *-*, zn.
For reasons of motivation we sacrifice details in the remainder of
this section, clarifying any misconceptions beginning with $5.2.
We consider differential forms of class 00 with complex values on a
complex manifold. Let U be a coordinate neighborhood with (complex)