On the homotopy types of compact kaehler and complex projective manifolds

Abstract

We show that in every dimension greater than or equal to 4, there exist compact Kaehler manifolds which do not have the homotopy type of projective complex manifolds. Thus they a fortiori are not deformation equivalent to a projective manifold, which solves negatively Kodaira's problem. We give both non simply connected (of dimension at least 4) and simply connected (of dimension at least 6) such examples.

To appear in Inventiones Math

Cited by in corpus (22)

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• Formality, Alexander invariants, and a question of Serre
• Some aspects of explicit birational geometry inspired by complex dynamics
• Riemannian Holonomy and Algebraic Geometry
• Algebraic approximation of Kähler threefolds of Kodaira dimension zero
• The fundamental group of compact K{ä}hler threefolds
• Algebraic approximations of compact Kähler threefolds
• Torsion points on the cohomology jump loci of compact Kähler manifolds
• Characteristic varieties and logarithmic differential 1-forms
• Kaehler structures on spin 6-manifolds
• Twisted cscK metrics and Kähler slope stability
• Perspectives on geometric analysis
• Algebraic approximations of compact Kähler manifolds of algebraic codimension 1
• Bimeromorphic automorphism groups of non-projective hyperkähler manifolds - a note inspired by C. T. McMullen
• The Kodaira problem for Kähler spaces with vanishing first Chern class
• Kaehler groups and duality
• Multiplicative sub-Hodge structures of conjugate varieties
• Homomorphisms between Kaehler groups
• Dimension four simply connected Voisin manifolds
• Algebraic approximation of Kähler threefolds
• Projectivity of Kaehler manifolds - Kodaira's problem (after C. Voisin)
• Geometric structures, the Gromov order, Kodaira dimensions and simplicial volume