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We extend the decomposition theorem for numerically $K$-trivial varietieswith log terminal singularities to the K\"ahler setting. Along the way we provethat all such varieties admit a strong locally trivial algebraic approximation,thus completing the numerically $K$-trivial case of a conjecture of Campana andPeternell.

Algebraic approximation and the decomposition theorem for K\"ahler Calabi-Yau varieties