Compact Kähler manifolds of algebraic dimension zero are isogenous to products of Kummer and simple manifolds; four-dimensional strictly simple ones are étale tori quotients or holomorphically symplectic.
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Bogomolov decomposition and compact K{\"a}hler manifolds of algebraic dimension zero
Compact Kähler manifolds of algebraic dimension zero are isogenous to products of Kummer and simple manifolds; four-dimensional strictly simple ones are étale tori quotients or holomorphically symplectic.