Fibrations of compact Kähler manifolds in terms of cohomological properties of their fundamental groups (Q1977473)

If \(X\) denotes a compact Kähler manifold whose fundamental group is \(\Gamma\), the author considers some fibration theorems with conditions on the first cohomology groups of \(\Gamma\) with respect to unitary representations in complex Hilbert spaces. Thus, if for some unitary representation \(\Phi\) of \(\Gamma\) the first cohomology group \(H^{1}(\Gamma, \Phi)\) does not vanish, then either \(X\) is of general type or for some unramified finite cover of \(X\) there exists a holomorphic fibration of \(X\) over a complex compact torus or for some nonsingular Kähler modification of \(X\), there exists a holomorphic fibration \(\sigma\) with positive-dimensional fibers over a projective manifold of logarithmic general type with respect to the multiplicity locus of \(\sigma\). The author's method allows to obtain also some other results among which an interesting result on algebraic dimensions of compact Kähler manifolds.