Algebraic approximation of Kähler threefolds with Kodaira dimension zero
We prove that every compact Kähler threefold with canonical singularities and vanishing first Chern class admits an algebraic approximation, i.e. a flat deformation containing projective fibres arbitrarily close to the central fibre. Combined with recent progress in the Kähler MMP, this implies that any compact Kähler manifold of dimension three and Kodaira dimension zero admits an algebraic approximation after a suitable bimeromorphic modification. In particular, the fundamental group of such a Kähler manifold is almost abelian.
In the course of the proof, we show that for a canonical threefold with first Chern class zero, the Albanese map is a surjective analytic fibre bundle with connected fibre. This generalizes a result of Kawamata (valid in all dimensions) to the Kähler case. Furthermore we generalize a criterion for algebraic approximability, due to Green and Voisin, to quotients of a manifold by a finite group.