On generalized Inoue manifolds

Abstract

This paper is about a generalization of famous Inoue's surfaces. Let $M$ be a matrix in $SL(2n+1,\mathbb{Z})$ having only one real eigenvalue which is simple. We associate to $M$ a complex manifold $T_M$ of complex dimension $n+1$. This manifold fibers over $S^1$ with the fiber $\mathbb{T}^{2n+1}$ and monodromy $M^\top$. Our construction is elementary and does not use algebraic number theory. We show that some of the Oeljeklaus-Toma manifolds are biholomorphic to the manifolds of type $T_M$. We prove that if $M$ is not diagonalizable, then $T_M$ does not admit a K\"ahler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds.