Complex tori (Q5894639)

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 [\textit{F. Oort} and \textit{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 \textit{D. Bertrand} [Duke Math. J. 80, No. 1, 223--250 (1995; Zbl 0847.11036)].