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

TL;DR

This paper demonstrates that compact K"ahler threefolds with Kodaira dimension 0 or 1 can be approximated algebraically through specific bimeromorphic models, advancing understanding of their deformation properties.

Contribution

It establishes the existence of algebraic approximations for K"ahler threefolds with Kodaira dimension 0 or 1 via new bimeromorphic models with controlled singularities.

Findings

Existence of $Q$-factorial bimeromorphic models with terminal singularities.

Local triviality of algebraic approximations around curves.

Every such threefold admits an algebraic approximation.

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.