Complex torus

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