On second non-HLC degree of closed symplectic manifold

Abstract

In this note, we show that for a closed almost-Kähler 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.

arXiv comment10 pages, appeared in Int. J. Math. We have corrected the typos of the title of the original document in IJM

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