View source for Complex torus

{{TEX|done}}
{{MSC|22|57Sxx}}

A complex torus is 
a complex Abelian Lie group obtained from the $n$-dimensional complex space $\C^n$ by factorizing with respect to a lattice $\def\G{\Gamma}\G\subset \C^n $ of rank $2n$. Every connected compact complex Lie group is a complex torus
{{Cite|Mu}}. Every Hermitian scalar product in $\C^n$ defines on $T=\C^n/\G$ a translation-invariant Kähler metric. Complex tori can also be characterized as the only compact parallelizable Kähler manifolds
{{Cite|Wa}}. The group of automorphisms of the complex manifold $T$ is the same as the holomorph of the group $T$ as a complex Lie group (cf.
[[Holomorph of a group|Holomorph of a group]]).

Holomorphic $p$-forms on a complex torus $T$ have the form

$$\sum_{i_1<\cdots<i_p} a_{i_1\dots i_p} dz_{i_1}\wedge\cdots\wedge dz_{i_p}\;,$$
where $a_{i_1\cdots i_p}\in \C$, $z_1,\dots,z_n$ are the coordinates in $C^n$ and the Dolbeault cohomology ring $\sum_{p,q=0}^n\; H^{p,q}(T)$ is naturally isomorphic to $\wedge {C^n}^*\otimes \wedge\overline{{C^n}^*}$ (see
{{Cite|Mu}}).

As real Lie groups, all $n$-dimensional complex tori are $2n$-dimensional tori and are isomorphic for fixed $n$. From the point of view of their complex structure their behaviour is extremely complicated. A basis for a lattice $\G\subset \C^n$ can be given by a matrix $\def\O{\Omega}\O$ of dimension $n\times 2n$, called the period matrix of the torus $T=\C^n/\G$. Tori $T_i = \C^n/\G_i$ with period matrices $\O_i$ ($i=1,2$) are isomorphic (as complex Lie groups or as complex manifolds) if and only if there exist matrices $C\in \textrm{GL}(n,\C) $ and $Z\in \textrm{GL}(2n,Z)$ such that $\O_2 = C\O_1 Z$.

The period matrix of an $n$-dimensional torus can be reduced to the form $\|EA\|$, where $\textrm{Im} |A| > 0 $. Tori with matrices of this form generate a holomorphic family that gives an effectively-parametrized versal deformation of any $n$-dimensional complex torus depending on $n^2$ parameters . In particular, for $n=1$, the parameter space is the upper half-plane $\textrm{Im}\; a >0$, and the set of isomorphism classes of one-dimensional complex tori can be identified with the quotient $\{\textrm{Im}\; a > 0 \}/\Delta$ where $\Delta$ is the
[[Modular group|modular group]].

Complex tori that are algebraic varieties are called Abelian varieties (cf.
[[Abelian variety|Abelian variety]]). A complex torus $\C^n/\G$ is an Abelian variety if and only there exists in $\C^n$ a Hermitian scalar product whose imaginary part is integer-valued on $\G\times \G$
{{Cite|Mu}}. In terms of the period matrix this is the Riemann–Frobenius condition: There should exist a skew-symmetric matrix $Q\in \textrm{GL}(2n,Z)$ such that $\O Q\O' = 0$ and $-i\O Q\overline{\O'}$ is positive definite. When $n=1$ this condition always holds; the corresponding algebraic curves are elliptic (cf.
[[Elliptic curve|Elliptic curve]]). The period matrix

$$\O=\begin{pmatrix}1&0&i\sqrt{2}&i\sqrt{5}\\
0&1&i\sqrt{3}&i\sqrt{7} \end{pmatrix}$$
provides an example of a two-dimensional complex torus that is not an algebraic variety. On this torus there are not even non-constant meromorphic functions
{{Cite|Si}}. A necessary and sufficient condition that an $n$-dimensional complex torus be algebraic is the existence on it of $n$ algebraically-independent meromorphic functions.

Interest in complex tori originated in the 19th century in connection with the study of Abelian functions (cf.
[[Abelian function|Abelian function]]) and Jacobi varieties of algebraic curves (cf.
[[Jacobi variety|Jacobi variety]]). To any $n$-dimensional compact Kähler manifold $M$ there is related a collection of $n$ complex tori, its intermediate Jacobi varieties
{{Cite|Ch}}.

====References====
{|
|-
|valign="top"|{{Ref|Ch}}||valign="top"|  S.S. Chern,   "Complex manifolds without potential theory", Springer  (1979)  {{MR|0533884}}  {{ZBL|0444.32004}}                
|-
|valign="top"|{{Ref|KoSp}}||valign="top"|  K. Kodaira,   D.C. Spencer,   "On deformations of complex analytic structures I"  ''Ann. of Math.'', '''67'''  (1958)  pp. 328–400  {{MR|0112154}}  {{ZBL|0128.16901}}                
|-
|valign="top"|{{Ref|KoSp2}}||valign="top"|  K. Kodaira,   D.C. Spencer,   "On deformations of complex analytic structures II"  ''Ann. of Math.'', '''67'''  (1958)  pp. 403–466  {{MR|0112154}}  {{ZBL|0128.16901}}                
|-
|valign="top"|{{Ref|Mu}}||valign="top"|  D. Mumford,   "Abelian varieties", Oxford Univ. Press  (1974)   {{ZBL|0326.14012}}                
|-
|valign="top"|{{Ref|Si}}||valign="top"|  C.L. Siegel,   "Automorphe Funktionen in mehreren Variablen", Math. Inst. Göttingen  (1955)                  
|-
|valign="top"|{{Ref|Wa}}||valign="top"|  H.C. Wang,   "Complex parallisable manifolds"  ''Proc. Amer. Math. Soc.'', '''5'''  (1954)  pp. 771–776  {{MR|0074064}}                 
|-
|valign="top"|{{Ref|We}}||valign="top"|  A. Weil,   "Introduction à l'étude des variétés kählériennes", Hermann  (1958)    {{MR|0111056}}  {{ZBL|0137.41103}}    
|-
|valign="top"|{{Ref|We2}}||valign="top"|  R.O. Wells jr.,   "Differential analysis on complex manifolds", Springer  (1980)  {{MR|0608414}}  {{ZBL|0435.32004}}              
|-
|}