Symplectic non-Kähler manifolds with and without the Hard Lefschetz Condition

2026/06/22 by Richard Hind, Adriano Tomassini · 1 voice
Mathematics · #math.SG #math.DG

paper · pdf

Abstract

In this paper we construct compact manifolds without Kähler structures that admit both a symplectic form satisfying the Hard Lefschetz Condition (HLC) and another symplectic form that does not. Our construction builds upon the orbifold introduced by Fernández and Muñoz and its symplectic resolution studied by Cavalcanti, Fernández, and Muñoz. By considering a one-parameter family of symplectic forms on the orbifold, we show that the corresponding resolved manifolds fail to satisfy the HLC for all parameters. However, after performing a suitable symplectic blowup along a union of tori, we obtain a family of symplectic manifolds for which the HLC holds for all non-zero parameters but fails at the central parameter. As a consequence, we exhibit a smooth manifold with no Kähler structure whose space of symplectic forms contains both HLC and non-HLC structures in the same connected component. This provides new examples of the subtle interplay between symplectic topology and the Hard Lefschetz property.

Citations

• Symplectic solvmanifolds not satisfying the hard-Lefschetz condition
• Hard Lefschetz Condition on symplectic non-Kähler solvmanifolds
• Hard Lefschetz property of symplectic structures on compact Kaehler manifolds
• Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems
• Symplectic resolutions, Lefschetz property and formality
• An 8-dimensional non-formal simply connected symplectic manifold
• The Lefschetz property, formality and blowing up in symplectic geometry
• Formality of canonical symplectic complexes and Frobenius manifolds
• Three constructions of Frobenius manifolds: a comparative study

Discussions

• Richard Hind, Adriano Tomassini: Symplectic non-K\"ahler manifolds with and without the Hard Lefschetz Condition https://arxiv.org/abs/2606.23996 https://arxiv.org/pdf/2606.23996 https://arxiv.org/htm [bsky, 0 points, 0 comments]

Related

• Hard Lefschetz property of symplectic structures on compact Kaehler manifolds
• Hard Lefschetz Condition on symplectic non-Kähler solvmanifolds
• A 6-dimensional simply connected complex and symplectic manifold with no Kähler metric
• Formality and the Lefschetz property in symplectic and cosymplectic geometry
• Examples of Non-Kähler Hamiltonian circle manifolds with the strong Lefschetz property