The Bochner-Hartogs dichotomy for bounded geometry hyperbolic K\"ahler manifolds

TL;DR

This paper proves a dichotomy for hyperbolic K"ahler manifolds with bounded geometry and one end, showing either a vanishing cohomology or the existence of a holomorphic map to a Riemann surface.

Contribution

It establishes a Bochner-Hartogs type dichotomy for a class of hyperbolic K"ahler manifolds with bounded geometry and one end, linking cohomology vanishing to the existence of holomorphic fibrations.

Findings

Either the first compactly supported cohomology vanishes or the manifold admits a proper holomorphic map to a Riemann surface.

The result applies to connected hyperbolic complete K"ahler manifolds with bounded geometry of order two and one end.

Provides a new criterion connecting topological and complex-analytic properties of such manifolds.

Abstract

The main result is that for a connected hyperbolic complete K\"ahler manifold with bounded geometry of order two and exactly one end, either the first compactly supported cohomology with values in the structure sheaf vanishes or the manifold admits a proper holomorphic mapping onto a Riemann surface.