The fundamental group of compact Kähler threefolds

Abstract

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