Complex torus
In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number N circles). Here N must be the even number 2n, where n is the complex dimension of M.

All such complex structures can be obtained as follows: take a lattice Λ in a vector space V isomorphic to Cn considered as real vector space; then the quotient group

is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For n = 1 this is the classical period lattice construction of elliptic curves. For n > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties.

The actual projective embeddings are complicated (see equations defining abelian varieties) when n > 1, and are really coextensive with the theory of theta-functions of several complex variables (with fixed modulus). There is nothing as simple as the cubic curve description for n = 1. Computer algebra can handle cases for small n reasonably well. By Chow's theorem, no complex torus other than the abelian varieties can 'fit' into projective space.

Definition 
One way to define complex tori is as a compact connected complex Lie group . These are Lie groups where the structure maps are holomorphic maps of complex manifolds. It turns out that all such compact connected Lie groups are commutative, and are isomorphic to a quotient of their Lie algebra  whose covering map is the exponential map of a Lie algebra to its associated Lie group. The kernel of this map is a lattice  and .

Conversely, given a complex vector space  and a lattice  of maximal rank, the quotient complex manifold  has a complex Lie group structure, and is also compact and connected. This implies the two definitions for complex tori are equivalent.

Period matrix of a complex torus 
One way to describe a complex toruspg 9 is by using a  matrix  whose columns correspond to a basis  of the lattice  expanded out using a basis  of . That is, we writesoWe can then write the torus  asIf we go in the reverse direction by selecting a matrix , it corresponds to a period matrix if and only if the corresponding matrix  constructed by adjoining the complex conjugate matrix  to , sois nonsingular. This guarantees the column vectors of  span a lattice in  hence must be linearly independent vectors over .

Example 
For a two-dimensional complex torus, it has a period matrix of the formfor example, the matrixforms a period matrix since the associated period matrix has determinant 4.

Normalized period matrix 
For any complex torus  of dimension  it has a period matrix  of the formwhere  is the identity matrix and  where . We can get this from taking a change of basis of the vector space  giving a block matrix of the form above. The condition for  follows from looking at the corresponding -matrixsince this must be a non-singular matrix. This is because if we calculate the determinant of the block matrix, this is simplywhich gives the implication.

Example 
For example, we can write a normalized period matrix for a 2-dimensional complex torus asone such example is the normalized period matrixsince the determinant of  is nonzero, equal to .

Period matrices of Abelian varieties 
To get a period matrix which gives a projective complex manifold, hence an algebraic variety, the period matrix needs to further satisfy the Riemann bilinear relations.

Homomorphisms of complex tori 
If we have complex tori  and  of dimensions  then a homomorphismpg 11 of complex tori is a functionsuch that the group structure is preserved. This has a number of consequences, such as every homomorphism induces a map of their covering spaceswhich is compatible with their covering maps. Furthermore, because  induces a group homomorphism, it must restrict to a morphism of the latticesIn particular, there are injections and which are called the analytic and rational representations of the space of homomorphisms. These are useful to determining some information about the endomorphism ring  which has rational dimension .

Holomorphic maps of complex tori 
The class of homomorphic maps between complex tori have a very simple structure. Of course, every homomorphism induces a holomorphic map, but every holomorphic map is the composition of a special kind of holomorphic map with a homomorphism. For an element  we define the translation map sending Then, if  is a holomorphic map between complex tori , there is a unique homomorphism  such thatshowing the holomorphic maps are not much larger than the set of homomorphisms of complex tori.

Isogenies 
One distinct class of homomorphisms of complex tori are called isogenies. These are endomorphisms of complex tori with a non-zero kernel. For example, if we let  be an integer, then there is an associated map sending which has kernelisomorphic to .

Isomorphic complex tori 
There is an isomorphism of complex structures on the real vector space  and the setand isomorphic tori can be given by a change of basis of their lattices, hence a matrix in . This gives the set of isomorphism classes of complex tori of dimension , , as the Double coset spaceNote that as a real manifold, this has dimensionthis is important when considering the dimensions of moduli of Abelian varieties, which shows there are far more complex tori than Abelian varieties.

Line bundles and automorphic forms 
For complex manifolds , in particular complex tori, there is a constructionpg 571 relating the holomorphic line bundles  whose pullback  are trivial using the group cohomology of . Fortunately for complex tori, every complex line bundle  becomes trivial since .

Factors of automorphy 
Starting from the first group cohomology groupwe recall how its elements can be represented. Since  acts on  there is an induced action on all of its sheaves, hence onThe -action can then be repsented as a holomorphic map . This map satisfies the cocycle condition iffor every  and . The abelian group of 1-cocycles  is called the group of factors of automorphy. Note that such functions  are also just called factors.

On complex tori 
For complex tori, these functions  are given by functionswhich follow the cocycle condition. These are automorphic functions, more precisely, the automorphic functions used in the transformation laws for theta functions. Also, any such map can be written asforwhich is useful for computing invariants related to the associated line bundle.

Line bundles from factors of automorphy 
Given a factor of automorphy  we can define a line bundle on  as follows: the trivial line bundle  has a -action given byfor the factor . Since this action is free and properly discontinuous, the quotient bundleis a complex manifold. Furthermore, the projection  induced from the covering projection . This gives a mapwhich induces an isomorphismgiving the desired result.

For complex tori 
In the case of complex tori, we have  hence there is an isomorphismrepresenting line bundles on complex tori as 1-cocyles in the associated group cohomology. It is typical to write down the group  as the lattice  defining , hencecontains the isomorphism classes of line bundles on .

First chern class of line bundles on complex tori 
From the exponential exact sequencethe connecting morphismis the first Chern class map, sending an isomorphism class of a line bundle to its associated first Chern class. It turns out there is an isomorphism between  and the module of alternating forms on the lattice , . Therefore,  can be considered as an alternating -valued 2-form  on . If  has factor of automorphy  then the alternating form can be expressed asfor  and .

Example 
For a normalized period matrixexpanded using the standard basis of  we have the column vectors defining the lattice . Then, any alternating form  on  is of the formwhere a number of compatibility conditions must be satisfied.

Sections of line bundles and theta functions 
For a line bundle  given by a factor of automorphy , so  and , there is an associated sheaf of sections  wherewith  open. Then, evaluated on global sections, this is the set of holomorphic functions  such thatwhich are exactly the theta functions on the plane. Conversely, this process can be done backwards where the automorphic factor in the theta function is in fact the factor of automorphy defining a line bundle on a complex torus.

Hermitian forms and the Appell-Humbert theorem 

For the alternating -valued 2-form  associated to the line bundle , it can be extended to be -valued. Then, it turns out any -valued alternating form  satisfying the following conditions

 
  for any 

is the extension of some first Chern class  of a line bundle . Moreover, there is an associated Hermitian form  satisfying

 
 

for any .

Neron-Severi group 

For a complex torus  we can define the Neron-Serveri group  as the group of Hermitian forms  on  withEquivalently, it is the image of the homomorphismfrom the first Chern class. We can also identify it with the group of alternating real-valued alternating forms  on  such that .

Example of a Hermitian form on an elliptic curve 
For an elliptic curve  given by the lattice  where  we can find the integral form  by looking at a generic alternating matrix and finding the correct compatibility conditions for it to behave as expected. If we use the standard basis  of as a real vector space (so ), then we can write out an alternating matrix

and calculate the associated products on the vectors associated to . These are

Then, taking the inner products (with the standard inner product) of these vectors with the vectors  we get

so if , then

We can then directly verify , which holds for the matrix above. For a fixed , we will write the integral form as . Then, there is an associated Hermitian formgiven by where

Semi-character pairs for Hermitian forms 
For a Hermitian form  a semi-character is a mapsuch thathence the map  behaves like a character twisted by the Hermitian form. Note that if  is the zero element in , so it corresponds to the trivial line bundle , then the associated semi-characters are the group of characters on . It will turn out this corresponds to the group  of degree  line bundles on , or equivalently, its dual torus, which can be seen by computing the group of characterswhose elements can be factored as mapsshowing a character is of the formfor some fixed dual lattice vector . This gives the isomorphismof the set of characters with a real torus. The set of all pairs of semi-characters and their associated Hermitian form , or semi-character pairs, forms a group  whereThis group structure comes from applying the previous commutation law for semi-characters to the new semicharacter :It turns out this group surjects onto  and has kernel , giving a short exact sequenceThis surjection can be constructed through associating to every semi-character pair a line bundle .

Semi-character pairs and line bundles 
For a semi-character pair  we can construct a 1-cocycle  on  as a mapdefined asThe cocycle relationcan be easily verified by direct computation. Hence the cocycle determines a line bundlewhere the -action on  is given byNote this action can be used to show the sections of the line bundle  are given by the theta functions with factor of automorphy . Sometimes, this is called the canonical factor of automorphy for . Note that because every line bundle  has an associated Hermitian form , and a semi-character can be constructed using the factor of automorphy for , we get a surjectionMoreover, this is a group homomorphism with a trivial kernel. These facts can all be summarized in the following commutative diagramwhere the vertical arrows are isomorphisms, or equality. This diagram is typically called the Appell-Humbert theorem.

Dual complex torus 

As mentioned before, a character on the lattice can be expressed as a functionfor some fixed dual vector . If we want to put a complex structure on the real torus of all characters, we need to start with a complex vector space which  embeds into. It turns out that the complex vector spaceof complex antilinear maps, is isomorphic to the real dual vector space , which is part of the factorization for writing down characters. Furthermore, there is an associated latticecalled the dual lattice of . Then, we can form the dual complex toruswhich has the special property that that dual of the dual complex torus is the original complex torus. Moreover, from the discussion above, we can identify the dual complex torus with the Picard group of by sending an anti-linear dual vector  togiving the mapwhich factors through the dual complex torus. There are other constructions of the dual complex torus using techniques from the theory of Abelian varietiespg 123-125. Essentially, taking a line bundle  over a complex torus (or Abelian variety) , there is a closed subset  of  defined as the points of  where their translations are invariant, i.e.Then, the dual complex torus can be constructed aspresenting it as an isogeny. It can be shown that defining  this way satisfied the universal properties of , hence is in fact the dual complex torus (or Abelian variety).

Poincare bundle 
From the construction of the dual complex torus, it is suggested there should exist a line bundle  over the product of the torus  and its dual which can be used to present all isomorphism classes of degree 0 line bundles on . We can encode this behavior with the following two properties

  for any point  giving the line bundle 
  is a trivial line bundle

where the first is the property discussed above, and the second acts as a normalization property. We can construct  using the following hermitian formand the semi-characterfor . Showing this data constructs a line bundle with the desired properties follows from looking at the associated canonical factor of , and observing its behavior at various restrictions.

See also
Poincare bundle
Complex Lie group
Automorphic function
Intermediate Jacobian
Elliptic gamma function

References

Complex 2-dimensional tori 
 is an Abelian Surface Isomorphic or Isogeneous to a Product of Elliptic Curves? - Gives tools to find complex tori which are not Abelian varieties
 surfaces and products of elliptic curves

Gerbes on complex tori 

Gerbes and the Holomorphic Brauer Group of Complex Tori - Extends idea of using alternating forms on the lattice to , to construct gerbes on a complex torus
Mukai duality for gerbes with connection - includes examples of gerbes on complex tori
Equivariant gerbes on complex tori
A Gerbe for the Elliptic Gamma Function - could be extended to complex tori

P-adic tori 

 p-adic Abelian Integrals: from Theory to Practice

Complex manifolds
Complex surfaces
Abelian varieties