On generalized Inoue manifolds

Abstract

This paper is about a generalization of famous Inoue's surfaces. Let M be a matrix in SLp2n `1, Zq 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 T 2n`1 and monodromy M J . 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 Kähler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds.

On a generalization of Inoue and Oeljeklaus-Toma manifolds

“…, α s q and A consists of just one matrix U J . For the case s " 1 this action is nothing else than the one constructed in our previous paper [6]. For the case s ą 1 this action is not non-degenerate.…”

“…For s " 1 the family consisting of one polynomial D 1 ptq " t is obviously a Dirichlet family for any matrix M of type J 0 . Thus we recover here the construction of generalized Inoue manifolds from [6], §2.…”

“…The ring Zrts{C has natural projections φ j : Zrts{C Ñ Zrts{B j . Consider the product of these projections (6) φ : Zrts{C Ñ Zrts{B 0 ˆZrts{B 1 ˆ... ˆZrts{B k .…”

“…, D s pM J q. [6]. In this subsection we explain the relation of manifolds T pM, Dq with the generalized Inoue manifolds, constructed in [6].…”

“…The paper of D. Angella, M. Parton, and V. Vuletescu [1] is dedicated to the proof of the rigidity of the complex structure of the OT-manifolds. The non-existence of complex curves in OTmanifolds is proved in the paper [21] of S. Verbitsky. In the paper [6] we introduced another generalization of the Inoue surfaces. Namely for every matrix M of type I we constructed a complex non-Kähler manifold T M that fibers over a circle and the monodromy of the fibration equals M J .…”