Algebraic approximation of K\"ahler threefolds of Kodaira dimension zero

TL;DR

This paper demonstrates that K"ahler threefolds with Kodaira dimension zero can be approximated by projective varieties through small deformations, extending known results and revealing new structural properties.

Contribution

It proves the density of projective fibers in the deformation space of such threefolds and generalizes key results to the K"ahler setting, including Albanese map decomposition and algebraic approximability criteria.

Findings

Projective fibers are dense in the deformation space of K"ahler threefolds with Kodaira dimension zero.

Every such threefold admits small projective deformations after a bimeromorphic modification.

The fundamental group of any K"ahler threefold relates to those of projective manifolds.

Abstract

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.