Wikipedia / Complex torus
Wikipedia / 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.
has a complex Lie group structure, and is also compact and connected. This implies that the two definitions for complex tori are equivalent.
Period matrix of a complex torus
One way to describe a g-dimensional complex torus is nonsingular.
Example
forms a period matrix since the associated period matrix has determinant 4.
Normalized period matrix
is the identity matrix. We can get this from taking a change of basis giving a block matrix which gives the implication.
Example
.
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
, then a homomorphism induces a group homomorphism.
Holomorphic maps of complex tori
showing the holomorphic maps are not much larger than the set of homomorphisms of complex tori.
Isogenies
.
Isomorphic complex tori
this is important when considering the dimensions of moduli of Abelian varieties, which shows that there are far more complex tori than Abelian varieties.
Line bundles and automorphic forms
, in particular complex tori, there is a construction relating the holomorphic line bundles are trivial using the group cohomology.
Factors of automorphy
. The abelian group is called the group of factors of automorphy are also just called factors.
On complex tori
which follow the cocycle condition. These are automorphic functions, more precisely, the automorphic functions used in the transformation laws for theta functions which is useful for computing invariants related to the associated line bundle.
Line bundles from factors of automorphy
giving the desired result.
For complex tori
.
First chern class of line bundles on complex tori
From the exponential exact sequence is the first Chern class.
Example
where a number of compatibility conditions must be satisfied.
Sections of line bundles and theta functions
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
satisfying the following conditions
satisfying
.
Néron-Severi group
we can define the Néron-Serveri group.
Example of a Hermitian form on an elliptic curve
For
Semi-character pairs for Hermitian forms
behaves like a character, then the associated semi-characters are the group of characters, or semi-character pairs.
Semi-character pairs and line bundles
. Sometimes, this is called the canonical factor of automorphy where the vertical arrows are isomorphisms, or equality. This diagram is typically called the Appell-Humbert theorem.
Dual complex torus
of complex antilinear. Then, we can form the dual complex torus which factors through the dual complex torus. 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).
Poincaré bundle
. We can encode this behavior with the following two properties
is a trivial line bundle
, and observing its behavior at various restrictions.
See also
References
Complex 2-dimensional tori
- Gives tools to find complex tori which are not Abelian varieties
Gerbes on complex tori
, to construct gerbes on a complex torus
- includes examples of gerbes on complex tori
- could be extended to complex tori
P-adic tori