Weakly special filtered ends of complete Kähler manifolds, proper holomorphic mappings to Riemann surfaces, and the Bochner–Hartogs dichotomy

Results concerning the structure of connected noncompact complete Kähler manifolds with (filtered) ends of one of several possible weakly special types are considered. For example, an end for which there is a nonnegative continuous plurisubharmonic function on the manifold that is unbounded on the end and for which the manifold has bounded geometry along each level in the end is weakly special of type (LI). In particular, every end of a connected covering manifold of a connected noncompact weakly 1-complete complete Kähler manifold is of type (LI). It is shown that a connected noncompact complete Kähler manifold that admits a weakly special ends decomposition and has at least three weakly special filtered ends admits a proper holomorphic mapping onto a Riemann surface. It is also shown that the Bochner–Hartogs dichotomy holds for any one-ended connected complete Kähler with a weakly special end of type (LI); that is, either the first compactly supported cohomology with values in the structure sheaf vanishes, or there exists a proper holomorphic mapping onto a Riemann surface. From this it follows that the Bochner–Hartogs dichotomy holds for any one-ended connected covering space of a connected noncompact weakly 1-complete Kähler manifold. These results are obtained as applications of a version of Gromov’s cup product lemma that the authors had obtained earlier and that appeared elsewhere.