Complex torus - Imported Knowlpedia Content

Definition

A complex torus of complex dimension $g$ is a quotient

X=V/\Lambda,

where $V$ is a $g$ -dimensional complex vector space and $\Lambda\subset V$ , viewed as a real vector space, is a discrete subgroup of rank $2g$ whose real span is $V$ . Translation by $\Lambda$ acts freely and properly discontinuously, so the quotient is a compact complex manifold. Addition on $V$ descends to $X$ , making it a connected compact complex Lie group. Conversely, every connected compact complex Lie group is isomorphic to such a quotient. The lattice is part of a presentation, not additional chosen data on the abstract torus.

Geometry and topology

As a real smooth manifold, every $g$ -dimensional complex torus is diffeomorphic to $(S^1)^{2g}$ . A translation-invariant Hermitian inner product on $V$ descends to a flat Kähler metric on $X$ . Translation-invariant holomorphic one-forms also descend and trivialize the holomorphic cotangent bundle.

These properties follow directly from the quotient presentation and are summarized in Birkenhake–Lange, Chapter 1.

Examples and algebraicity

For $g=1$ , a complex torus is an elliptic curve with the identity as base point; analytically it has the form $\mathbb C/(\mathbb Z+\tau\mathbb Z)$ with $\operatorname{Im}\tau>0$ . In higher dimensions, not every complex torus is projective. A projective complex torus is an abelian variety, and projectivity is equivalent to the existence of a suitable positive integral $(1,1)$ -class, or Riemann form. Thus “complex torus” and “abelian variety” are not synonyms.

References

Christina Birkenhake and Herbert Lange, Complex Tori, Progress in Mathematics 177, Birkhäuser, 1999. Springer DOI record. Relevant: Chapter 1, quotient construction and basic geometry.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Springer DOI record. Relevant: the discussion of complex tori and Kähler manifolds.