On formality of some symplectic manifolds (Q2778012)

From author's introduction: If \(({\mathcal M},\kappa)\) is a symplectic \(2n\)-manifold and \(d^*\) is its symplectic codifferential operator, then \((\wedge^* ({\mathcal M}),d^*,d)\) is a differentiable Gerstenhaber-Batalin-Vilkovisky (dGBV) algebra, which is integrable (i.e., the \(dd^*\)-lemma holds) if and only if \({\mathcal M}\) satisfies the hard Lefschetz condition (HLC); that is, NEWLINE\[NEWLINE[\kappa ]^p:H^{n-p}({\mathcal M})\to H^{n+p}({\mathcal M}),\quad 0\leq p\leq n,NEWLINE\]NEWLINE is an isomorphism. In this paper we show that if a compact symplectic manifold \(({\mathcal M},\kappa)\) admits a \(\kappa\)-calibrated (almost) complex structure that is not too nonintegrable, then the HLC holds; moreover, Section 5.3 shows that the apparently brute force estimate we use is actually sharp.