Page 201 - Curvature and Homology

If f: M+ M' and g: M' + M" are holomorphic maps, so is the
composed map g f : M + M". By a holomorphic isomorphism f : M + M'
is meant a 1-1 holomorphic map f together with a differentiable
map g : M' -t M such that both f g and g f are the identity maps on
M' and M, respectively. Iff is a holomorphic isomorphism, it follows
that the inverse map g is also a holomorphic isomorphism.
Lemma 5.8.1. Let M be a complex manifold and f a complex-valued
dflerentiable function on M. In order that f be a holomorphic map of M
into C (considered as a complex manifold), it is necessary and suflcient that
f be a holomorphic function.
Since dz is a base for the forms of bidegree (1,O) on C, in order that f
be a holomorphic map, it is necessary and sufficient that f*(d2) = df
be of bidegree (1,O). Hence, since df = d'f + d"f, it is necessary and
sufficient that df'f vanish.

Lemma 5.8.2. The induced dual map of a holomorphic map sends holo-
morphic f m into hlomorphic form.
Let f: M -+ M' be a holomorphic map and a a form of bidegree
(p, 0) on M'. Then, since f* preserves bidegrees, f*(a) is a form of
bidegree (p, 0) on M. Hence, since f * and d commute, so do fr and d".
Thus, if a is holomorphic, so is f*(a).

Proposition 5.8.3. Let i@ be a cwering space of the complcs matu~old M
and IT the canonical projection of l@ onto M. ( We denote this covering space
by (I@, IT).) Then, there exists a uniqu complex structure on l@ with respect
to which IT is a holomorphic map.
For, let {Val be an open covering of l@ such that for every a the
restriction sr, of IT to Va is a homeomorphism of Va onto v(Va). Such
a covering of A? always exists. To each a is associated a complex structure
operator Ja on Va in terms of which IT, : Va + M is holomorphic. To
see this, we need only define ra,- J, = J. wa, . On the intersection Va n Vp,
the complex structure operators Ja and Jp coincide since wB-l IT, is the
identity map on Va n VB, and as such is holomorphic. Thus, the
operator on Icl having the Ja as its restrictions defines a complex
structure on M. With resptct to this complex structure on l@ the
projection IT is evidently a holomorphic map. The uniqueness is clear.

Corollary. Let (I@, n) be a cwering space of the Kaehler manifold M.
Then, (.a, IT) has a canonically dejined Kaehler structure.
For, let IR be the Kaehler 2-form of M canonically defined by the
Kaehler metric dF2 of M. Let IT* denote the induced dual map of IT.