  We prove that any compact K\"ahler 3-dimensional manifold which has no
non-trivial complex subvarieties is a torus. This is a very special case of a
general conjecture on the structure of 'simple manifolds', central in the
bimeromorphic classification of compact K\"ahler 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 K\"ahler 3-dimensional analytic space with terminal singularities and nef
canonical bundle is a cyclic quotient of a simple non-projective torus if it
carries no effective divisor. This is a crucial step to complete the
bimeromorphic classification of compact K\"ahler 3-folds
