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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 $\Cal{H}_{\omega}^k(M)$ of all harmonic $k$-forms on $M$ and the symplectic harmonic $k$-th cohomology group $H_{\omega\text{-}hr}^k(M):=\Cal{H}_{\omega}^k(M)/B^k(M)\cap\Cal{H}_{\omega}^k(M)$ [see {\it 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 $\frak{g}$ 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 $\Cal{H}_{\omega}^k(\frak{g})$ be the space of all $G$-invariant harmonic $k$-forms on $G/\Gamma$ and $H_{\omega\text{-}hr}^k(\frak{g}):=\Cal{H}_{\omega}^k(\frak{g})/B^k(\frak{g})\cap\Cal{H}_{\omega}^k(\frak{g})$. The authors prove that $B^3(\frak{g})\subset \Cal{H}_{\omega}^3(\frak{g})$ and some properties concerning $\dim H_{\omega\text{-}hr}^{2m-2}(\frak{g})$ and $\dim H_{\omega\text{-}hr}^{2m-1}(\frak{g})$. In particular, the dimension of $H_{\omega\text{-}hr}^{2m-1}(\frak{g})$ 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 $G/\Gamma\ (m\ge 6)$ admitting symplectic structures $\omega$ such that the dimension of $H_{\omega\text{-}hr}^3(\frak{g})$ varies [see also {\it D. Yan}, Adv. Math. 120, 143--154 (1996; Zbl 0872.58002)\rbrack, for a question raised by Khesin and McDuff]. For the entire collection see [Zbl 1008.00022].
53D35Global theory of symplectic and contact manifolds
57R17Symplectic and contact topology