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Algebraic approximation and the decomposition theorem for Kähler Calabi-Yau varieties

We extend the decomposition theorem for numerically $K$-trivial varieties with log terminal singularities to the Kähler setting. Along the way we prove that all such varieties admit a strong locally trivial algebraic approximation, thus completing the numerically $K$-trivial case of a conjecture of Campana and Peternell.