Compact Kahler 3-manifolds without nontrivial subvarieties
We prove that any compact Kahler 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 classication of compact Kahler manifolds. The proof follows from the Brunella pseudo-eectivity theorem, combined with fundamental results of Siu and of the second author on the Le- long numbers of closed positive (1;1)-currents, and with a version of the hard Lefschetz theorem for pseudo-eective line bundles, due to Takegoshi and Demailly-Peternell- Schneider. In a similar vein, we show that a normal compact and Kahler 3-dimensional analytic space with terminal singularities and nef canonical bundle is a cyclic quotient of a simple nonprojective torus if it carries no eective divisor. This is a crucial step towards completing the bimeromorphic classication of compact Kahler threefolds.