Implications of Complex Structure
Since holomorphic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds.
For example, the Whitney embedding theorem tells us that every smooth manifold can be embedded as a smooth submanifold of Rn, whereas it is "rare" for a complex manifold to have a holomorphic embedding into Cn. Consider for example any compact connected complex manifold M: any holomorphic function on it is locally constant by Liouville's theorem. Now if we had a holomorphic embedding of M into Cn, then the coordinate functions of Cn 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 Cn are called Stein manifolds 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 structures, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the 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) Cn gives an orientation, as biholomorphic maps are orientation-preserving).
Read more about this topic: Complex Manifold
Famous quotes containing the words implications of, implications, complex and/or structure:
“The power to guess the unseen from the seen, to trace the implications of things, to judge the whole piece by the pattern, the condition of feeling life in general so completely that you are well on your way to knowing any particular corner of it—this cluster of gifts may almost be said to constitute experience.”
—Henry James (1843–1916)
“When it had long since outgrown his purely medical implications and become a world movement which penetrated into every field of science and every domain of the intellect: literature, the history of art, religion and prehistory; mythology, folklore, pedagogy, and what not.”
—Thomas Mann (1875–1955)
“The human mind is so complex and things are so tangled up with each other that, to explain a blade of straw, one would have to take to pieces an entire universe.... A definition is a sack of flour compressed into a thimble.”
—Rémy De Gourmont (1858–1915)
“The verbal poetical texture of Shakespeare is the greatest the world has known, and is immensely superior to the structure of his plays as plays. With Shakespeare it is the metaphor that is the thing, not the play.”
—Vladimir Nabokov (1899–1977)