Symplectic cohomologies and deformations

In this note we study the behavior of symplectic cohomology groups under symplectic deformations. Moreover, we show that for compact almost-K"ahler manifolds $(X,J,g,\omega )$ with $J$ ${\mathcal{C}}^{\mathrm{\infty }}$-pure and full the space of de Rham harmonic forms is contained in the space of symplectic-Bott-Chern harmonic forms. Furthermore, we prove that the second non-HLC degree measures the gap between the de Rham and the symplectic-Bott-Chern harmonic forms.