Dimension four simply connected Voisin manifolds

TL;DR

This paper proves that certain simply connected compact Kähler manifolds in dimension four, constructed by Voisin and Oguiso, do not share the rational homotopy type of complex projective manifolds, extending previous results to lower dimensions.

Contribution

It extends Voisin's and Oguiso's constructions by showing their examples in dimension four lack the rational homotopy type of complex projective manifolds.

Findings

Voisin's dimension four examples lack the rational homotopy type of projective manifolds.

Oguiso's examples cannot deform into projective manifolds.

The results extend previous higher-dimensional findings to dimension four.

Abstract

Voisin constructed a series of examples concerning simply connected compact K\"ahler manifolds of even dimensions, which do not have the rational homotopy type of a complex projective manifold starting from dimension six. In this note, we prove that Voisin's examples of dimension four also does not have the rational homotopy type of a complex projective manifold. Oguiso constructed simply connected compact K\"ahler manifolds starting from dimension four, which can not deform to a complex projective manifold under a small deformation. We also prove that Oguiso's examples do not have the rational homotopy type of a complex projective manifold.