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A complex Abelian Lie group obtained from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c0241901.png" />-dimensional complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c0241902.png" /> by factorizing with respect to a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c0241903.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c0241904.png" />. Every connected compact complex Lie group is a complex torus [[#References|[1]]]. Every Hermitian scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c0241905.png" /> defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c0241906.png" /> a translation-invariant Kähler metric. Complex tori can also be characterized as the only compact parallelizable Kähler manifolds [[#References|[2]]]. The group of automorphisms of the complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c0241907.png" /> is the same as the holomorph of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c0241908.png" /> as a complex Lie group (cf. [[Holomorph of a group|Holomorph of a group]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c0241909.png" />-forms on a complex torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419010.png" /> have the form
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Holomorphic $p$-forms on a complex torus $T$ have the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419011.png" /></td> </tr></table>
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$$\sum_{i_1<\cdots<i_p} a_{i_1\dots i_p} dz_{i_1}\wedge\cdots\wedge dz_{i_p}\;,$$
 +
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}}).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419013.png" /> are the coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419014.png" /> and the Dolbeault cohomology ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419015.png" /> is naturally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419016.png" /> (see [[#References|[1]]]).
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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 $\O_2 = C\O_1 Z$.
  
As real Lie groups, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419017.png" />-dimensional complex tori are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419018.png" />-dimensional tori and are isomorphic for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419019.png" />. From the point of view of their complex structure their behaviour is extremely complicated. A basis for a lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419020.png" /> can be given by a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419021.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419022.png" />, called the period matrix of the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419023.png" />. Tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419024.png" /> with period matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419025.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419026.png" /> are isomorphic (as complex Lie groups or as complex manifolds) if and only if there exist matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419029.png" />.
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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
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[[Modular group|modular group]].
  
The period matrix of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419030.png" />-dimensional torus can be reduced to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419032.png" />. Tori with matrices of this form generate a holomorphic family that gives an effectively-parametrized versal deformation of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419033.png" />-dimensional complex torus depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419034.png" /> parameters . In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419035.png" />, the parameter space is the upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419036.png" />, and the set of isomorphism classes of one-dimensional complex tori can be identified with the quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419037.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419038.png" /> is the [[Modular group|modular group]].
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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
  
Complex tori that are algebraic varieties are called Abelian varieties (cf. [[Abelian variety|Abelian variety]]). A complex torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419039.png" /> is an Abelian variety if and only there exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419040.png" /> a Hermitian scalar product whose imaginary part is integer-valued on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419041.png" /> [[#References|[1]]]. In terms of the period matrix this is the Riemann–Frobenius condition: There should exist a skew-symmetric matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419042.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419044.png" /> is positive definite. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419045.png" /> this condition always holds; the corresponding algebraic curves are elliptic (cf. [[Elliptic curve|Elliptic curve]]). The period matrix
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$$\O=\begin{pmatrix}1&0&i\sqrt{2}&i\sqrt{5}\\
 +
0&1&i\sqrt{3}&i\sqrt{7} \end{pmatrix}$$
 +
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419046.png" /></td> </tr></table>
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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.
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 [[#References|[5]]]. A necessary and sufficient condition that an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419047.png" />-dimensional complex torus be algebraic is the existence on it of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419048.png" /> algebraically-independent meromorphic functions.
+
[[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}}.
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419049.png" />-dimensional compact Kähler manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419050.png" /> there is related a collection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024190/c02419051.png" /> complex tori, its intermediate Jacobi varieties [[#References|[7]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford,  "Abelian varieties" , Oxford Univ. Press  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.C. Wang,  "Complex parallisable manifolds" ''Proc. Amer. Math. Soc.'' , '''5''' (1954pp. 771–776</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top"> K. Kodaira,  D.C. Spencer,  "On deformations of complex analytic structures I"  ''Ann. of Math.'' , '''67'''  (1958)  pp. 328–400</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top"> K. Kodaira,  D.C. Spencer,  "On deformations of complex analytic structures II"  ''Ann. of Math.'' , '''67'''  (1958)  pp. 403–466</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Weil,  "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> C.L. Siegel,  "Automorphe Funktionen in mehrerer Variablen" , Math. Inst. Göttingen  (1955)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R.O. Wells jr.,  "Differential analysis on complex manifolds" , Springer (1980)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S.S. Chern,  "Complex manifolds without potential theory" , Springer  (1979)</TD></TR></table>
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{|
 +
|-
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|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}}   
 +
|-
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|valign="top"|{{Ref|We2}}||valign="top"| R.O. Wells jr.,  "Differential analysis on complex manifolds", Springer  (1980) {{MR|0608414}}  {{ZBL|0435.32004}}             
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|-
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|}

Revision as of 21:15, 3 March 2012

2020 Mathematics Subject Classification: Primary: 22-XX Secondary: 57Sxx [MSN][ZBL]

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 [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 [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).

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 [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.

Complex tori that are algebraic varieties are called Abelian varieties (cf. 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 $\G\times \G$ [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). 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 [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) and Jacobi varieties of algebraic curves (cf. Jacobi variety). To any $n$-dimensional compact Kähler manifold $M$ there is related a collection of $n$ complex tori, its intermediate Jacobi varieties [Ch].

References

[Ch] S.S. Chern, "Complex manifolds without potential theory", Springer (1979) MR0533884 Zbl 0444.32004
[KoSp] K. Kodaira, D.C. Spencer, "On deformations of complex analytic structures I" Ann. of Math., 67 (1958) pp. 328–400 MR0112154 Zbl 0128.16901
[KoSp2] K. Kodaira, D.C. Spencer, "On deformations of complex analytic structures II" Ann. of Math., 67 (1958) pp. 403–466 MR0112154 Zbl 0128.16901
[Mu] D. Mumford, "Abelian varieties", Oxford Univ. Press (1974) Zbl 0326.14012
[Si] C.L. Siegel, "Automorphe Funktionen in mehreren Variablen", Math. Inst. Göttingen (1955)
[Wa] H.C. Wang, "Complex parallisable manifolds" Proc. Amer. Math. Soc., 5 (1954) pp. 771–776 MR0074064
[We] A. Weil, "Introduction à l'étude des variétés kählériennes", Hermann (1958) MR0111056 Zbl 0137.41103
[We2] R.O. Wells jr., "Differential analysis on complex manifolds", Springer (1980) MR0608414 Zbl 0435.32004
How to Cite This Entry:
Complex torus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complex_torus&oldid=21460
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article