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.

Definition

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 torus

One way to describe a g-dimensional complex torus[2]: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 .

Example

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 matrix

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.

Example

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 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.[3]

Homomorphisms of complex tori

If we have complex tori and of dimensions then a homomorphism[2]: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 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 that

showing 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 kernel

isomorphic to .

Isomorphic complex tori

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 forms

For complex manifolds , in particular complex tori, there is a construction[2]: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 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 represented 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 tori

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 automorphy

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 tori

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 tori

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 .

Example

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 functions

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 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

  1. 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 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 curve

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 forms

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 bundles

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 torus

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]: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 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

  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 also

References

  1. Mumford, David (2008). Abelian varieties. C. P. Ramanujam, I︠U︡. I. Manin. Published for the Tata Institute of Fundamental Research. ISBN 978-8185931869. OCLC 297809496.
  2. 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".
  • 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 tori

Gerbes on complex tori

P-adic tori

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