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.

The complex torus associated to a lattice spanned by two periods, ω1 and ω2. Corresponding edges are identified.

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.

DefinitionEdit

One way to define complex tori[1] 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 torusEdit

One way to describe a complex torus[2]pg 9 is by using a   matrix   whose columns correspond to a basis   of the lattice   expanded out using a basis   of  . That is, we write

 
so
 
We can then write the torus   as
 
If 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  , so
 
is nonsingular. This guarantees the column vectors of   span a lattice in   hence must be linearly independent vectors over  .

ExampleEdit

For a two-dimensional complex torus, it has a period matrix of the form

 
for example, the matrix
 
forms a period matrix since the associated period matrix has determinant 4.

Normalized period matrixEdit

For any complex torus   of dimension   it has a period matrix   of the form

 
where   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  -matrix
 
since this must be a non-singular matrix. This is because if we calculate the determinant of the block matrix, this is simply
 
which gives the implication.

ExampleEdit

For example, we can write a normalized period matrix for a 2-dimensional complex torus as

 
one such example is the normalized period matrix
 
since the determinant of   is nonzero, equal to  .

Period matrices of Abelian varietiesEdit

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.[3]

Homomorphisms of complex toriEdit

If we have complex tori   and   of dimensions   then a homomorphism[2]pg 11 of complex tori is a function

 
such that the group structure is preserved. This has a number of consequences, such as every homomorphism induces a map of their covering spaces
 
which is compatible with their covering maps. Furthermore, because   induces a group homomorphism, it must restrict to a morphism of the lattices
 
In 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 toriEdit

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 that
 
showing the holomorphic maps are not much larger than the set of homomorphisms of complex tori.

IsogeniesEdit

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 kernel
 
isomorphic to  .

Isomorphic complex toriEdit

There is an isomorphism of complex structures on the real vector space   and the set

 
and 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 space
 
Note that as a real manifold, this has dimension
 
this 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 formsEdit

For complex manifolds  , in particular complex tori, there is a construction[2]pg 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 automorphyEdit

Starting from the first group cohomology group

 
we recall how its elements can be represented. Since   acts on   there is an induced action on all of its sheaves, hence on
 
The  -action can then be repsented as a holomorphic map  . This map satisfies the cocycle condition if
 
for 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 toriEdit

For complex tori, these functions   are given by functions

 
which 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 as
 
for
 
which is useful for computing invariants related to the associated line bundle.

Line bundles from factors of automorphyEdit

Given a factor of automorphy   we can define a line bundle on   as follows: the trivial line bundle   has a  -action given by

 
for the factor  . Since this action is free and properly discontinuous, the quotient bundle
 
is a complex manifold. Furthermore, the projection   induced from the covering projection  . This gives a map
 
which induces an isomorphism
 
giving the desired result.

For complex toriEdit

In the case of complex tori, we have   hence there is an isomorphism

 
representing 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  , hence
 
contains the isomorphism classes of line bundles on  .

First chern class of line bundles on complex toriEdit

From the exponential exact sequence

 
the connecting morphism
 
is 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 as
 
for   and  .
ExampleEdit

For a normalized period matrix

 
expanded using the standard basis of   we have the column vectors defining the lattice  . Then, any alternating form   on   is of the form
 
where a number of compatibility conditions must be satisfied.

Sections of line bundles and theta functionsEdit

For a line bundle   given by a factor of automorphy  , so   and  , there is an associated sheaf of sections   where

 
with   open. Then, evaluated on global sections, this is the set of holomorphic functions   such that
 
which 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 theoremEdit

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

  1.  
  2.   for any  

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

  1.  
  2.  

for any  .

Neron-Severi groupEdit

For a complex torus   we can define the Neron-Serveri group   as the group of Hermitian forms   on   with

 
Equivalently, it is the image of the homomorphism
 
from 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 curveEdit

For[4] 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 form
 
given by
 
where  

Semi-character pairs for Hermitian formsEdit

For a Hermitian form   a semi-character is a map

 
such that
 
hence 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 characters

 

whose elements can be factored as maps

 

showing a character is of the form

 

for some fixed dual lattice vector  . This gives the isomorphism

 

of 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   where

 
This 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 sequence
 
This surjection can be constructed through associating to every semi-character pair a line bundle  .

Semi-character pairs and line bundlesEdit

For a semi-character pair   we can construct a 1-cocycle   on   as a map

 
defined as
 
The cocycle relation
 
can be easily verified by direct computation. Hence the cocycle determines a line bundle
 
where the  -action on   is given by
 
Note 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 surjection
 
Moreover, this is a group homomorphism with a trivial kernel. These facts can all be summarized in the following commutative diagram
 
where the vertical arrows are isomorphisms, or equality. This diagram is typically called the Appell-Humbert theorem.

Dual complex torusEdit

As mentioned before, a character on the lattice can be expressed as a function

 

for 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 space

 

of 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 lattice

 

called the dual lattice of  . Then, we can form the dual complex torus

 

which 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   to

 

giving the map

 

which factors through the dual complex torus. There are other constructions of the dual complex torus using techniques from the theory of Abelian varieties[1]pg 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 as

 

presenting 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 bundleEdit

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

  1.   for any point   giving the line bundle  
  2.   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 form

 

and the semi-character

 

for  . 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 alsoEdit

ReferencesEdit

  1. ^ a b Mumford, David (2008). Abelian varieties. C. P. Ramanujam, I︠U︡. I. Manin. Published for the Tata Institute of Fundamental Research. ISBN 8185931860. OCLC 297809496.
  2. ^ a b c Birkenhake, Christina (2004). Complex Abelian Varieties. Herbert Lange (Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-662-06307-1. OCLC 851380558.
  3. ^ "Riemann bilinear relations" (PDF). Archived (PDF) from the original on 31 May 2021.
  4. ^ "How Appell-Humbert theorem works in the simplest case of an elliptic curve".{{cite web}}: CS1 maint: url-status (link)
  • Birkenhake, Christina; Lange, Herbert (1999), Complex tori, Progress in Mathematics, vol. 177, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4103-0, MR 1713785

Complex 2-dimensional toriEdit

Gerbes on complex toriEdit

P-adic toriEdit