Bogomolov decomposition and compact K{\"a}hler manifolds of algebraic dimension zero

We prove conditionally that compact K\''ahler manifolds of algebraic dimension zero are (essentially) isogeneous to products of Kummer and `simple' ones, the latter being conjecturally bimeromorphically symplectic. `Simple' means: its general point is not contained in a nontrivial subvariety.

We also prove that four-dimensional `strictly simple' manifolds are either étale quotients of tori or holomorphically symplectic. `Strictly simple' means: its only subvarieties are points and itself.