On second non-HLC degree of closed symplectic manifold

TL;DR

This paper investigates the relationship between de Rham harmonic forms and symplectic-Bott-Chern harmonic forms on closed almost-Kähler manifolds, especially in four dimensions, revealing conditions under which these spaces coincide or differ.

Contribution

It establishes a link between the second non-HLC degree and the gap between harmonic form spaces on certain symplectic manifolds, providing new insights into their structure.

Findings

Harmonic forms are contained within symplectic-Bott-Chern harmonic forms under specific conditions.

In four-dimensional manifolds with $b_2^+=1$, the second non-HLC degree quantifies the difference between the two harmonic form spaces.

The paper characterizes the second non-HLC degree in terms of harmonic form spaces on closed almost-Kähler manifolds.

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.