On Symplectically Harmonic Forms on Six-dimensional Nilmanifolds

TL;DR

This paper investigates how the dimensions of symplectically harmonic cohomology classes vary on 6-dimensional nilmanifolds with different symplectic forms, providing new examples of such variation and answering a question posed by Khesin and McDuff.

Contribution

It describes the variation of symplectically harmonic cohomology dimensions on 6-dimensional nilmanifolds and shows that these dimensions can vary with the symplectic form, unlike in 4-dimensional cases.

Findings

Certain 6-dimensional nilmanifolds have families of symplectic forms with varying harmonic cohomology dimensions.

The variation of these dimensions affirms a question by Khesin and McDuff.

In contrast, 4-dimensional nilmanifolds do not exhibit such variation.

Abstract

In the present paper we study the variation of the dimensions $h_k$ of spaces of symplectically harmonic cohomology classes (in the sense of Brylinski) on closed symplectic manifolds. We give a description of such variation for all 6-dimensional nilmanifolds equipped with symplectic forms. In particular, it turns out that certain 6-dimensional nilmanifolds possess families of homogeneous symplectic forms $\omega_t$ for which numbers $h_k(M,\omega_t)$ vary with respect to t. This gives an affirmative answer to a question raised by Boris Khesin and Dusa McDuff. Our result is in contrast with the case of 4-dimensional nilmanifolds which do not admit such variations by a remark of Dong Yan.