The fundamental group of compact K{\"a}hler threefolds

TL;DR

This paper proves that the fundamental group of any compact Kähler threefold matches that of some projective manifold and establishes algebraic approximations for certain Kähler manifolds, advancing the understanding of their structure.

Contribution

It demonstrates the existence of a projective manifold with the same fundamental group as a given Kähler threefold and proves algebraic approximations for specific Kähler manifolds, resolving key aspects of the bimeromorphic Kodaira problem.

Findings

Fundamental groups of compact Kähler threefolds are realizable by projective manifolds.

Established algebraic approximations for Kähler manifolds of algebraic dimension one less than their complex dimension.

Resolved the bimeromorphic Kodaira problem for compact Kähler threefolds.

Abstract

Let $X$ be a compact K{\"a}hler manifold of dimension three. We prove that there exists a projective manifold $Y$ such that $\pi\_1(X)\simeq \pi\_1(Y)$. We also prove the bimeromorphic existence of algebraic approximations for compact K{\"a}hler manifolds of algebraic dimension $\dim(X)-1$. Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact K{\"a}hler threefolds.