On formality of some symplectic manifolds

Let (M, κ) be a compact symplectic manifold of dimension 2n; we prove that, if there exists a κ-calibrated complex structure J such that, with respect to the Hermitian structure gJ, the Nijenhuis tensor AJ satisfies, for $\mathbf0\boldsymbol<\boldsymbol p\boldsymbol<\boldsymbol n\boldsymbol,\textbf{?}{\boldsymbol\Arrowvert\mathbf A\boldsymbol\Arrowvert}\boldsymbol<\boldsymbol K{\boldsymbol(\mathbf n\boldsymbol,\mathbf p\boldsymbol)}\sqrt{{\mathbf\mu}_{\mathbf r\boldsymbol+\mathbf1}{\boldsymbol(\mathbf J\boldsymbol)}}$ (where r = n − p and r (J) is the smallest positive eigenvalue of Δg J, acting on r + 1-forms), then (M, κ) satisfies the hard Lefschetz condition. Note that this implies that the differentiable Gerstenhaber-Batalin-Vilkovisky algebra $(\mathbf\Lambda^{\ast}{(\mathbf M)},\mathbf d^{\star},\mathbf d)$ is integrable (i.e., the $\boldsymbol d\boldsymbol d^{\star}$-lemma holds). Consequently, the formality of the deRham complexand the existence of a structure of (formal) Frobenius manifold on C [[H ^* (M)]] follow at once; an example shows that the given estimate is sharp.