Oort, Frans; Zarhin, Yuri G. Complex tori. (English) Zbl 0879.14023 Indag. Math., New Ser. 7, No. 4, 473-487 (1996). The main objects of the paper under review are complex tori \(X\) and their first homology groups \(\Lambda (X)=H_1(X,\mathbb Z)\). The goal is to describe the category of complex tori in module-theoretic terms. The authors construct a ring \(F(X)\) which is an order in the Hodge algebra of \(X\) introduced in their previous work [F. Oort and Yu. G. Zarhin, Math. Ann. 303, No. 1, 11–29 (1995; Zbl 0858.14024)]. They show that the category of complex subtori of \(X\) is equivalent to the category of \(F(X)\)-submodules \(\Gamma\) of \(\Lambda (X)\) with torsion-free quotient \(\Lambda (X)/\Gamma\). They construct a three-dimensional complex torus with infinitely many non-isomorphic (and even non-isogenous) subtori; an analogous example of a semi-abelian variety with infinitely many non-isomorphic semi-abelian varieties was independently constructed by D. Bertrand [Duke Math. J. 80, No. 1, 223–250 (1995; Zbl 0847.11036)]. Reviewer: Boris Kunyavskii (Ramat Gan) Cited in 2 Documents MSC: 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 32J99 Compact analytic spaces Keywords:complex torus; first homology groups; Hodge algebra; non-isomorphic subtori Citations:Zbl 0847.11036; Zbl 0858.14024 PDFBibTeX XMLCite \textit{F. Oort} and \textit{Y. G. Zarhin}, Indag. Math., New Ser. 7, No. 4, 473--487 (1996; Zbl 0879.14023) Full Text: DOI References: [1] Bertrand, D., Minimal heights and polarizations on group varieties, Duke Math. J., 80, 223-250 (1995) · Zbl 0847.11036 [2] Lang, S., Complex multiplication, (Grundl. Math. Wiss., 255 (1983), Springer-Verlag) · Zbl 0536.14029 [3] Lenstra, Jr, H.W., F. Oort and Yu.G. Zarhin — Abelian subvarieties. J. Algebra, to appear.; Lenstra, Jr, H.W., F. Oort and Yu.G. Zarhin — Abelian subvarieties. J. Algebra, to appear. [4] Mumford, D., A note of Shimura’s paper ‘Discontinuous groups and abelian varieties’, Math. Ann., 181, 345-351 (1969) · Zbl 0169.23301 [5] Oort, F.; Zarhin, Yu. G., Endomorphism algebras of complex tori, Math. Ann., 303, 11-29 (1995) · Zbl 0858.14024 [6] Shafarevich, I. R., Basic algebraic geometry, (Grundl., Bd. 213 (1974), Springer-Verlag) · Zbl 1082.14501 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.