On second non-HLC degree of closed symplectic manifold

Abstract

In this note, we show that for a closed almost-K\"{a}hler manifold $(X,J)$ with the almost complex structure $J$ satisfies $\dim\ker P_{J}=b_{2}-1$ the space of de Rham harmonic forms is contained in the space of symplectic-Bott-Chern harmonic forms. In particular, suppose that $X$ is four-dimension, if the self-dual Betti number $b^{+}_{2}=1$, then we prove that the second non-HLC degree measures the gap between the de Rham and the symplectic-Bott-Chern harmonic forms.