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.