Complex Tori

On an even-dimensional complex torus there are many symplectic structures. All of them are uniquely determined by their values at a certain point. It is therefore reasonable first to investigate separately the question of the existence of foliation into tori of smaller dimensions and then to find out with respect to which symplectic structure this foliation is isotropic. It turns out that on a general complex torus there are no foliations into tori of smaller dimensions, and if they do exist, they are locally trivial. 

PROOF Let t r — Af be an embedding of tori, t an induced mapping of universal coverings, and t C — C . Then the Jacobian matrix of t is a periodic holomorphic function on with a complete lattice of periods of rank 2p and is therefore constant. Consequently, t is an affine mapping, and we have come to the desired conclusion. 

Proposition 3.4.3. Let M be a manifold with a trivial canonical class and let f M N be a morphism of manifolds whose general Bbre is a torus. Then f cannot have simple multiple hbres. 

Proposition 3.4.4. For a general complex torus M of dimension m there exists not a single proper subtorus T C M. 

PROOF The torus M is uniquely defined by setting an integer-valued lattice Hz C (j2m 2m and an operator of complex structure on J Hu 

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Now if g > 2, most complex tori C9 /L have no non-constant meromorphic functions on them at all, and are not algebraic varieties, and do not carry any but trivial ea s27. In the case of a curve C, however, special things happen let s look for bilinear forms as candidates for B. We saw above that on R C) one has a positive definite Hermitian form ... 

The purpose of this lecture is to consider the map carrying C to its Jacobian Jac from a moduli point of view. Jac is a particular kind of complex torus and the Schottky problem is simply the problem of characterizing the complex tori that arise as Jacobians. The Torelli theorem says that Jac, plus the form H on its universal covering space, determine the curve C up to isomorphism. 

First of all, we saw that if g > 2, not all complex tori X = C9 /L are even projective varieties in fact, necessary and sufficient for X to be a projective variety is that there exists a positive definite Hermitian form H on O, such that E = Im H is integral on Lx L. The varieties that arise this way are called... 

An example of a manifold which is completely integrable (and even integrable without degeneracies) is given by the product of two nonalgebraic complex tori of equal dimension with a symplectic structure, in which multipliers are isotropic. But this manifold is not meromorphic ally integrable.. 

Theorem 3.4.5. Let M be a compact Kahlerian manifold of dimension m 4 and ci(M) = 0. Suppose that a morphism of comples manifolds f M N is given and that all Bbres of this morphism are p dimensional complex tori, where... 

Proof. First suppose is a complex torus T. Taking a basis of V, we may suppose we are in the above situation. (If some coordinates are 0, we replace P by a subspace.) We may also assume that Y is the unique closed orbit in the closure of x. The uniqueness follows from the existence of T -invariant polynomial which separates two disjoint T -invariant closed subsets (Theorem 3.3). 

In order to illustrate how the closedness of the orbit is determined, we consider the case Gc is the complex torus Tc = (C )r. We choose a basis aq,..., xn of V so that Tc is contained in the group of nonsingular diagonal matrices. Then we have distinguished... 

Theorem. The existence of a positive definite Hermitian H on C9 and an integral skew-symmetric E on L satisfying E = Im H is necessary and sufficient for a complex torus C9 /L to carry g algebraically independent meromorphic functions and if it has such functions, it admits an embedding into Pn, some n, hence is a projective variety29. 

Here we see the principle emerging that a complex torus does not fit easily in Pn non-trivial identities ( ) are required before it will fit at all. Now define a theta-function30 of order n to be an entire function / on C9 such that... 

An irreducible even-dimensional complex torus (see Example 1 below) may serve as an example of a manifold integrated additively, but not integrated in the weak sense. However, any symplectic form on a torus in some linear coordinates is written in the canonical form as... 

Hui = — 1. The exact meaning of the assertion of Proposition 3.4.4 is as follows. Let P be a smooth manifold of all complex structures J C End Hu (it is readily seen that they form a manifold). Then there exists an everywhere dense set P C P, such that for any complex structure J G P, a complex torus Mj = (Pr/Pi, J) does not have proper subtori. 

Corollary 3.4.1. On a generaP even-dimensional complex torus, there is not a single completely integrable simplectic structure. 

After a two-dimensional complex torus, the next simplest example of a compact complex symplectic manifold is a P3 type surface. 

Definition 3.4.6 A symplectic structure M w) will be called isotrivial if there exists a finite nonbranching covering x M — M, such that M splits into the direct product of an n-dimensional complex torus, completely isotropic with respect to the form... 

As a final example we consider noncovalent molecular complex formation with the macrocyclic ligand a-cyclodextrin, a natural product consisting of six a-D-glucose units linked 1-4 to form a torus whose cavity is capable of including molecules the size of an aromatic ring. Table 4-3 gives some rate constants for this reaction, where L represents the cyclodextrin and S is the substrate ... 

Spin trapping methods were also used to show that when carotenoid-P-cyclodextrin 1 1 inclusion complex is formed (Polyakov et al. 2004), cyclodextrin does not prevent the reaction of carotenoids with Fe3+ ions but does reduce their scavenging rate toward OOH radicals. This implies that different sites of the carotenoid interact with free radicals and the Fe3+ ions. Presumably, the OOH radical attacks only the cyclohexene ring of the carotenoid. This indicates that the torus-shaped cyclodextrins, Scheme 9.6, protects the incorporated carotenoids from reactive oxygen species. Since cyclodextrins are widely used as carriers and stabilizers of dietary carotenoids, this demonstrates a mechanism for their safe delivery to the cell membrane before reaction with oxygen species occurs. 

Empty ft- (11) and y-cyclodextrins (12) also take normal torus shapes, as revealed by X-ray crystallography, and no significant deformations are observed for these two cyclodextrins when they bind guest molecules. It is noteworthy that all empty cyclodextrins include water molecules in their cavities as shown in Table IV and Fig. 4. Since no water molecules were observed in inclusion complexes of cyclodextrins with organic guest molecules, it is evident that the expulsion of these water molecules in the cyclodextrin cavities is one of the important factors for formation of the inclusion complexes. 

Exercise 6.5 Use Proposition 6.3 to prove that every irreducible representation of the circle group T is one dimensional. Then generalize this result to prove that every irreducible representation of an n-fold product of circles T X X T (otherwise known as an n-torus) is one dimensional. (As always in this text, representations are complex vector spaces, so one dimensional refers to one complex dimension.)... 

If the quotient o>/a>0 is irrational, the path across the toroidal surface will return to a different point on the completion of each cycle. Eventually the trajectory will pass over every point on the surface of the torus without ever forming a closed loop. This is quasi-periodicity , and an example is shown in Fig. 13.11. The corresponding concentration histories do not necessarily give complex waveforms, as can be seen from the figure. However, the period of the oscillations is neither simply that of the natural cycle nor just that of the forcing term, but involves both. 

If the CFM is formed by a complex pair, the unit circle can be crossed at some point (actually at two points simultaneously) off the real axis. Generally this will correspond to the bifurcation from a stable limit cycle to a quasi-periodic motion on a torus. In special cases, however, where the crossing point corresponds to the fcth complex root of - 1, the limit cycle bifurcates to a phase-locked cycle (closed loop on the torus) corresponding to period-fc resonance such as that described ip 13.3. 

In a similar way we can prove that the embedded cell complex of the molecular (4,2)-torus link (see Figure 18) is topologically chiral. Also, by adding appropriate labels we can similarly prove the topological chirality of the oriented embedded cell complex of the molecular Hopf link (see Figure 19). 

As an illustration of the phenomena involved, consider the photocycloaddition of fumaronitrile to 5-X adamantanone, 6 (Scheme 2 X = F, Cl, Br, OH, Ph, or r-Bu) [94], In isotropic solvents, different quantities of adducts to the two carbonyl faces are formed. When 6 is complexed by /J-CD, the intrinsically more reactive face of the carbonyl group becomes more hindered toward attack by fumaronitrile than the less reactive one. As a result of this attractive interaction (NB, hydrogen bonding between the carbonyl oxygen and a hydroxyl on the fl-CD torus), the distribution of photoadducts is reversed. In this example, fl-CD serves the function of a reaction cavity with active walls (i.e., a template). 

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