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Complex tori. (English) Zbl 0879.14023

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)].

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
32J99 Compact analytic spaces
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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
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