Algebraic approximations of compact K\"ahler threefolds of Kodaira dimension 0 or 1

Abstract

We prove that every compact K\"ahler threefold $X$ of Kodaira dimension $\kappa = 0$ or $1$ has a $\mathbf{Q}$-factorial bimeromorphic model $X'$ with at worst terminal singularities such that for each curve $C \subset X'$, the pair $(X',C)$ admits a locally trivial algebraic approximation such that the restriction of the deformation of $X'$ to some neighborhood of $C$ is a trivial deformation. As an application, we prove that every compact K\"ahler threefold with $\kappa = 0$ or $1$ has an algebraic approximation.