Algebraic approximation of Kähler threefolds of Kodaira dimension zero

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We prove that for a compact K\"ahler threefold with canonical singularities and vanishing first Chern class, the projective fibres are dense in the semiuniversal deformation space. This implies that every K\"ahler threefold of Kodaira dimension zero admits small projective deformations after a suitable bimeromorphic modification. As a corollary, we see that the fundamental group of any K\"ahler threefold is a quotient of an extension of fundamental groups of projective manifolds, up to subgroups of finite index. In the course of the proof, we show that for a canonical threefold with $c_1 = 0$, the Albanese map decomposes as a product after a finite \'etale base change. This generalizes a result of Kawamata, valid in all dimensions, to the K\"ahler case. Furthermore we generalize a Hodge-theoretic criterion for algebraic approximability, due to Green and Voisin, to quotients of a manifold by a finite group.