Compact Kähler 3-manifolds without nontrivial subvarieties

Abstract

We prove that any compact Kähler 3-dimensional manifold which has no nontrivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of so-called simple manifolds, central in the bimeromorphic classification of compact Kähler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of Siu and of the second author on the Lelong numbers of closed positive (1, 1)-currents, and with a version of the hard Lefschetz theorem for pseudo-effective line bundles, due to Takegoshi and Demailly-Peternell- Schneider. In a similar vein, we show that a normal compact and Kähler 3-dimensional analytic space with terminal singularities and nef canonical bundle is a cyclic quotient of a simple nonprojective torus if it carries no effective divisor. This is a crucial step towards completing the bimeromorphic classification of compact Kähler threefolds.