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Harmonic forms on compact symplectic 2-step nilmanifolds. (English) Zbl 1033.53078
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 257-270 (2003).
If $$(M,\omega)$$ is a $$2m$$-dimensional symplectic manifold, one can define the space $$\mathcal{H}_{\omega}^k(M)$$ of all harmonic $$k$$-forms on $$M$$ and the symplectic harmonic $$k$$-th cohomology group $$H_{\omega\text{-}hr}^k(M):=\mathcal{H}_{\omega}^k(M)/B^k(M)\cap\mathcal{H}_{\omega}^k(M)$$ [see J. Brylinski, J. Differ. Geom. 28, 93–114 (1988; Zbl 0634.58029)]. Assume that $$M$$ is a $$2$$-step compact nilmanifold $$G/\Gamma$$ (i.e. $$G$$ is a simply connected $$2m$$-dimensional Lie group, whose Lie algebra $$\mathfrak{g}$$ is $$2$$-step nilpotent and $$\Gamma$$ is a discrete subgroup of $$G$$ such that $$G/\Gamma$$ is compact) and $$\omega$$ a $$G$$-invariant symplectic form on $$G/\Gamma$$. Let $$\mathcal{H}_{\omega}^k(\mathfrak{g})$$ be the space of all $$G$$-invariant harmonic $$k$$-forms on $$G/\Gamma$$ and $$H_{\omega\text{-}hr}^k(\mathfrak{g}):=\mathcal{H}_{\omega}^k(\mathfrak{g})/B^k(\mathfrak{g})\cap\mathcal{H}_{\omega}^k(\mathfrak{g})$$. The authors prove that $$B^3(\mathfrak{g})\subset \mathcal{H}_{\omega}^3(\mathfrak{g})$$ and some properties concerning $$\dim H_{\omega\text{-}hr}^{2m-2}(\mathfrak{g})$$ and $$\dim H_{\omega\text{-}hr}^{2m-1}(\mathfrak{g})$$. In particular, the dimension of $$H_{\omega\text{-}hr}^{2m-1}(\mathfrak{g})$$ does not depend on $$\omega$$ [see also the second author, Osaka J. Math. 39, 363–381 (2002; Zbl 1012.53076)]. Some examples are considered. Among them a $$2$$-step compact nilmanifold $$G/\Gamma\;(m\geq 6)$$ admitting symplectic structures $$\omega$$ such that the dimension of $$H_{\omega\text{-}hr}^3(\mathfrak{g})$$ varies [see also D. Yan, Adv. Math. 120, 143–154 (1996; Zbl 0872.58002)], for a question raised by Khesin and McDuff].
For the entire collection see [Zbl 1008.00022].

MSC:
 53D35 Global theory of symplectic and contact manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension