# Hard Lefschetz property and Moishezon manifolds

Cohomology ring of a closed manifold $M^{2n}$ is said to satisfy hard Lefschetz property if there exists an element $a\in H^2(M,\mathbb R)$ such that multiplication by $a^k$ yields an isomorphism $$H^{n-k}(M,\mathbb R)\to H^{n+k}(M,\mathbb R)$$ for all $k=1,\dots,n$.

By the hard Lefschetz theorem cohomology ring of any compact Kahler manifold $(M,\omega)$ has this property with $a=[\omega]$.

My question is: What are the examples of (necessarily non-Kahler) Moishezon (or Fujiki class C) manifolds with cohomology ring not satisfying hard Lefschetz property?

• mathoverflow.net/questions/136049/…. For almost Kahler manifolds, see archipel.uqam.ca/3755/1/D1973.pdf – user21574 Jul 18 '17 at 23:04
• The Lefschetz property is equivalent to the $dδ$-lemma: There is a paper, A New Construction of Symplectic Manifolds Author(s): Robert E. Gompf Source: Annals of Mathematics, Second Series, Vol. 142, No. 3 (Nov., 1995), pp. 527-595 – user21574 Jul 18 '17 at 23:19
• See Theorem 0.10, arxiv.org/pdf/1302.0524.pdf – user21574 Jul 18 '17 at 23:30
• @JarekKędra: A Moishezon manifold is a compact complex manifold, say of dimension $n$, whose field of meromorphic functions is isomorphic to the field of meromorphic functions of a complex projective variety of dimension $n$, as field extensions of the complex numbers. – Ben McKay Jul 19 '17 at 7:05
• Theorem 1.1 of this preprint can give better definition in Kahler current language via positivity theory , called Demailly's characterization of Moishezon manifolds, mat.jhu.edu/~shiffman/publications/… – user21574 Jul 19 '17 at 11:47