Compact Kähler manifolds of algebraic dimension zero are essentially isogeneous to products of Kummer and simple manifolds; four-dimensional strictly simple ones are étale quotients of tori 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 essentially isogeneous to products of Kummer and simple manifolds; four-dimensional strictly simple ones are étale quotients of tori or holomorphically symplectic.