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.
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
If we have complex tori and of dimensions then a homomorphismpg 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 .
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.
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
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 .
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 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 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 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
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 .
For a semi-character pair we can construct a 1-cocycle on as a map
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.
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 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 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).
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 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.