Fibrations of compact Kähler manifolds in terms of cohomological properties of their fundamental groups

Abstract

We prove fibration theorems on compact Kähler manifolds with conditions on first cohomology groups of fundamental groups with respect to unitary representations into Hilbert spaces. If the fundamental group T of compact Kähler manifold X violates Property (T) of Kazhdan’s, then H 1 (Gamma,Φ)≠0 for some unitary representation Φ. By our earlier work there exists a d-closed holomorphic 1-form with coefficients twisted by some unitary representation Φ ′ , possibly non-isomorphic to Φ. Taking norms we obtains a positive semi-definite d-closed (1,1)-form ν sur x, which underlies a semi-Khäler structure. We study meromorphic foliations related to this semi-Khäler structure and another semi-Khäler structure related to the Ricci form to prove fibration theorems on some modification of an unramified finite cover of x. The base manifold is shown to be either a compact complex torus or a variety of logarithmic general type with respect to the multiplicity locus of the holomorphic fibration.