Holomorphic submersions of locally conformally Kähler manifolds (Q466947)

An almost Hermitian structure \((g,J,\omega)\) on a smooth manifold \(M\) is called locally conformal Kähler (lcK) if the almost complex structure \(J\) is integrable and \(g\) is locally conformal to a Kähler metric. The latter condition is equivalent to the existence of a closed 1-form \(\theta\), the Lee form, such that \(d\omega=\omega\wedge\theta\). Another equivalent definition of a lcK manifold is that of a complex manifold covered by a Kähler manifold, so that the deck transformations are homotheties. A lcK manifold is Vaisman if the Lee form is parallel with respect to the Levi-Civita connection. The paper under review studies lcK manifolds \(X\) which admit a holomorphic submersion to a complex manifold, with positive-dimensional fibers. The authors show that, if one of the fibers is of Kähler type, then either \(g\) is globally conformal to a Kähler metric, or \(X\) is biholomorphic to a finite quotient of a small deformation of a Vaisman manifold. The main result has an interesting application: The product of a compact lcK non-Kähler manifold with a compact Kähler manifold admits no lcK metric.