Symplectic cohomologies and deformations

TL;DR

This paper investigates how symplectic cohomology groups change under deformations and explores the relationship between harmonic forms in almost-Kähler manifolds, revealing new insights into their structure.

Contribution

It demonstrates the behavior of symplectic cohomology under deformations and establishes the inclusion of de Rham harmonic forms in symplectic-Bott-Chern harmonic forms for certain manifolds.

Findings

Symplectic cohomology groups vary predictably under symplectic deformations.

De Rham harmonic forms are contained within symplectic-Bott-Chern harmonic forms in specific almost-Kähler manifolds.

The second non-HLC degree quantifies the difference between de Rham and symplectic-Bott-Chern harmonic forms.

Abstract

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 $J$ $\mathcal{C}^\infty$-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.