Bogomolov decomposition and compact K{ä}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.