23-07-2012, 02:59 PM
Complex Manifolds and Deformation of Complex Structures
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Complex Manifolds
(a) Definition of Complex Manifolds
Recall that a Riemann surface ^ is a connected Hausdorff space S endowed
with a system of local complex coordinates {zi, 2 2 , . . . , Zj,...}. Each local
complex coordinate Zj is a homeomorphism Zji p^ Zj(p) of a domain Uj in
2 onto a domain % c: C such that Uj ^ = ^? ^i^<i that for each pair of
indices 7, /c with L/y n L4 ^ 0 , the map r^^: z^ip) ~> Zj{p), pe Ujn L4, is a
biholomorphic map from the open set ^^j ^ ^k onto ^ ^ c: ^^.. The concept
of a complex manifold is a natural generalization of the concept of a
Riemann surface. In case of a Riemann surface, the local complex coordinate
of a point p 6 S is a complex number. Using an n-tuple of complex numbers
Zj(p) = {zi(p),..., z„(p)) instead, we obtain the concept of an ndimensional
complex manifold. More precisely, let 2 be a connected Hausdorff
space, and {(7i,..., L^,...} an open covering of 1 consisting of at
most countably many domains. Suppose that on each ^ ^ 2, a homeomorphism
Differentiable Manifolds
A connected Hausdorff space S is called a topological manifold if there is
an open covering of 2 consisting of at most countably many domains
L/i,..., Uj,..., such that each Uj is homeomorphic to a domain % in R'".
In this case the homeomorphism of Uj onto %:
Compact Complex Manifolds
A complex manifold M is said to be compact if its underlying topological
manifold S is compact. In this book we mainly treat compact complex
manifolds.
Let M be a compact complex manifold. Then since M is covered by a
finite number of coordinate neighbourhoods, we may choose a system of
local cotnplex coordinates on M consisting of a finite number of local
coordinates { z i , . . . , z^v}. Let Uj be the domain of Zj: p-^ Zj{p), and put
Zj{Uj) = %c: C". We have M = [J. Uj. Identifying Uj with % as usual, we
may consider M = U^ %- Thus a compact complex manifold M is obtained
by glueing a finite number of domains ^ i , . . . , ^^^ inC" via the identification
o f Zfc G %L^j CZ OU. w i t h Zj = T,fc(Zfc) E %k C ^fc.