Algebraic approximation of Kähler threefolds of Kodaira dimension zero

Abstract

We prove that for a compact Kähler threefold with canonical singularities and vanishing first Chern class, the projective fibres are dense in the semiuniversal deformation space. This implies that every Kähler 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 Kähler 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 étale base change. This generalizes a result of Kawamata, valid in all dimensions, to the Kähler 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.

Final version. Significantly strengthened result on the Albanese map. To appear in Mathematische Annalen

Cited by in corpus (7)

• Equivariant Kuranishi family of complex compact manifolds
• Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups
• The fundamental group of compact K{ä}hler threefolds
• Algebraic approximations of compact Kähler manifolds of algebraic codimension 1
• Strictly nef divisors on singular threefolds
• The Kodaira problem for Kähler spaces with vanishing first Chern class
• A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four