The Seifert fiberings in the cancellation problem for elliptic curves (Q1070589)

Let \(T\) be a one-dimensional complex torus (an elliptic curve) without complex multiplication. It is known (T. Shioda) that if for two complex tori \(A\) and \(B\) the products \(A\times T\) and \(B\times T\) are isomorphic, then \(A\simeq B\). If however \(A\) and \(B\) are arbitrary compact analytic manifolds, then it does not, in general, follow from the isomorphism of products \(A\times T\) and \(B\times T\) that \(A\simeq B\) (a corresponding counter-example has been constructed by the second author [J. Reine Angew. Math. 322, 42--52 (1981; Zbl 0436.14001)]). In the paper under review, it is proved that if \(A\times T\simeq B\times T\) where \(A\) and \(B\) are compact analytic manifolds, then either \(A\simeq B\) or \(A\) and \(B\) are Seifert fiber bundles [\textit{H. Holmann}, Math. Ann. 157, 138--166 (1964; Zbl 0123.165)] over the same base space and with the same fibre, a complex torus which is isogenous to the torus \(T\). The formulation of this result does not remain to be true if the Seifert fiber bundles are replaced by locally trivial fiber bundles.