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

For n = 1 this is the classical period lattice construction of elliptic curves.

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).

By Chow's theorem, no complex torus other than the abelian varieties can 'fit' into projective space.

has a complex Lie group structure, and is also compact and connected.

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

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

This has a number of consequences, such as every homomorphism induces a map of their covering spaces

which are called the analytic and rational representations of the space of homomorphisms.

The class of homomorphic maps between complex tori have a very simple structure.

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.

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

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

, in particular complex tori, there is a construction[2]: 571  relating the holomorphic line bundles

representing line bundles on complex tori as 1-cocyles in the associated group cohomology.

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.

by looking at a generic alternating matrix and finding the correct compatibility conditions for it to behave as expected.

, or equivalently, its dual torus, which can be seen by computing the group of characters

This group structure comes from applying the previous commutation law for semi-characters to the new semicharacter

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

of complex antilinear maps, is isomorphic to the real dual vector space

Moreover, from the discussion above, we can identify the dual complex torus with the Picard group of

There are other constructions of the dual complex torus using techniques from the theory of Abelian varieties.

, hence is in fact the dual complex torus (or Abelian variety).

From the construction of the dual complex torus, it is suggested that there should exist a line bundle

and its dual which can be used to present all isomorphism classes of degree 0 line bundles on

Showing this data constructs a line bundle with the desired properties follows from looking at the associated canonical factor of