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
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
Example
For a two-dimensional complex torus, it has a period matrix of the form
Normalized period matrix
For any complex torus of dimension it has a period matrix of the form
Example
For example, we can write a normalized period matrix for a 2-dimensional complex torus as
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
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
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
Isomorphic complex tori
There is an isomorphism of complex structures on the real vector space and the set
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
On complex tori
For complex tori, these functions are given by functions
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 complex tori
In the case of complex tori, we have hence there is an isomorphism
First chern class of line bundles on complex tori
From the exponential exact sequence
Example
For a normalized period matrix
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
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
- 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
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
Semi-character pairs for Hermitian forms
For a Hermitian form a semi-character is a map such that
Semi-character pairs and line bundles
For a semi-character pair we can construct a 1-cocycle on as a map
Dual complex torus
As mentioned before, a character on the lattice can be expressed as a function
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
- 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
See also
- Poincare bundle
- Complex Lie group
- Automorphic function
- Intermediate Jacobian
- Elliptic gamma function
References
- ^ a b Mumford, David (2008). Abelian varieties. C. P. Ramanujam, I︠U︡. I. Manin. Published for the Tata Institute of Fundamental Research. ISBN 978-8185931869. OCLC 297809496.
- ^ 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.
- ^ "Riemann bilinear relations" (PDF). Archived (PDF) from the original on 31 May 2021.
- ^ "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
- Ruppert, Wolfgang M. (1990). "When is an abelian surface isomorphic or isogeneous to a product of elliptic curves?". Mathematische Zeitschrift. 203: 293–299. doi:10.1007/BF02570737. S2CID 120799085. - Gives tools to find complex tori which are not Abelian varieties
- Marchisio, Marina Rosanna (1998). "Abelian surfaces and products of elliptic curves". Bollettino dell'unione Matematica Italiana. 1-B (2): 407–427.
Gerbes on complex tori
- Ben-Bassat, Oren (2012). "Gerbes and the holomorphic Brauer group of complex tori". Journal of Noncommutative Geometry. 6 (3): 407–455. arXiv:0811.2746. doi:10.4171/JNCG/96. S2CID 15049025. - Extends idea of using alternating forms on the lattice to , to construct gerbes on a complex torus
- Block, Jonathan; Daenzer, Calder (2008). "Mukai duality for gerbes with connection". Crelle's Journal. arXiv:0803.1529v2. - includes examples of gerbes on complex tori
- Ben-Bassat, Oren (2013). "Equivariant gerbes on complex tori". Journal of Geometry and Physics. 64: 209–221. arXiv:1102.2312. Bibcode:2013JGP....64..209B. doi:10.1016/j.geomphys.2012.10.012. S2CID 119599648.
- Felder, Giovanni; Henriques, André; Rossi, Carlo A.; Zhu, Chenchang (2008). "A gerbe for the elliptic gamma function". Duke Mathematical Journal. 141. arXiv:math/0601337. doi:10.1215/S0012-7094-08-14111-0. S2CID 817920. - could be extended to complex tori
Recent Comments