Certain compact complex manifolds with infinite cyclic fundamental groups (Q1097379)

For each \({\mathbb{Z}}\)-linear transformation g of maximal rank of \({\mathbb{Z}}^ r\), \(r\geq 2\), admitting a real, simple, eigenvalue \(\lambda\) such that \(\lambda >| \eta |\) for all other eigenvalues, an r-dimensional toroidal compactification U with \(\pi_ 1(U)\cong {\mathbb{Z}}\) of the quotient of some open subset of \(({\mathbb{C}}^*)^ r\) by \(g^{{\mathbb{Z}}}\) is constructed; U hence contains an effective divisor X. When \(r=2\), U is either an hyperbolic or half Inoue surface. Some U's are shown to contain a global spherical shell. Many analytical invariants of U and the pair (U,X), in particular the deformations, are computed. Five explicit examples are given, with \(r=3\).