On Symplectically Harmonic Forms on Six-dimensional Nilmanifolds

Details On Symplectically Harmonic Forms on Six-dimensional Nilmanifolds On Symplectically Harmonic Forms on Six-dimensional Nilmanifolds

2000-02-07

arxiv.org/abs/math/0002053v1

Mathematics

Symplectic Geometry

Scientific paper

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.

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