On a generalization of Inoue and Oeljeklaus-Toma manifolds

In this paper we construct a family of complex analytic manifolds that generalize Inoue surfaces and Oeljeklaus-Toma manifolds. To a matrix $M$ in $SL(N,\mathbb{Z})$ satisfying some mild conditions on its characteristic polynomial we associate a manifold $T(M,\mathbf{D})$ (depending on an auxiliary parameter $\mathbf{D}$). This manifold fibers over the $s$-dimensional torus $\mathbb{T}^s$, where $s$ is the number of real eigenvalues of $M$. The fiber is the $N$-dimensional torus $\mathbb{T}^{N}$, and the monodromy matrices are certain polynomials of the matrix $M$. The basic difference of our construction from the preceding ones is that we admit non-diagonalizable matrices $M$ and the monodromy of the above fibration can also be non-diagonalizable. We prove that for a large class of non-diagonalizable matrices $M$ the manifold $T(M,\mathbf{D})$ does not admit any K\"ahler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds.