Editing Complex manifold (section)

==Implications of complex structure==
Since [[holomorphic function]]s are much more rigid than [[smooth function]]s, the theories of [[smooth manifold|smooth]] and complex manifolds have very different flavors: [[compact space|compact]] complex manifolds are much closer to [[algebraic variety|algebraic varieties]] than to differentiable manifolds.

For example, the [[Whitney embedding theorem]] tells us that every smooth ''n''-dimensional manifold can be [[Embedding|embedded]] as a smooth submanifold of '''R'''<sup>2''n''</sup>, whereas it is "rare" for a complex manifold to have a holomorphic embedding into '''C'''<sup>''n''</sup>. Consider for example any [[compact space|compact]] connected complex manifold ''M'': any holomorphic function on it is constant by [[Maximum modulus principle|the maximum modulus principle]]. Now if we had a holomorphic embedding of ''M'' into '''C'''<sup>''n''</sup>, then the coordinate functions of '''C'''<sup>''n''</sup> would restrict to nonconstant holomorphic functions on ''M'', contradicting compactness, except in the case that ''M'' is just a point. Complex manifolds that can be embedded in '''C'''<sup>''n''</sup> are called [[Stein manifold]]s and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties.

The classification of complex manifolds is much more subtle than that of differentiable manifolds. For example, while in dimensions other than four, a given topological manifold has at most finitely many [[smooth structure]]s, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. [[Riemann surface]]s, two dimensional manifolds equipped with a complex structure, which are topologically classified by the [[genus (mathematics)|genus]], are an important example of this phenomenon. The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a [[moduli space]], the structure of which remains an area of active research.

Since the transition maps between charts are biholomorphic, complex manifolds are, in particular, smooth and canonically oriented (not just [[orientable]]: a biholomorphic map to (a subset of) '''C'''<sup>''n''</sup> gives an orientation, as biholomorphic maps are orientation-preserving).