Harmonic forms on compact symplectic 2-step nilmanifolds. (English) Zbl 1033.53078

Sakane, Yusuke; Yamada, Takumi

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].

Reviewer: Mircea Craioveanu (Timişoara)

MSC:

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

Keywords:

2-step compact nilmanifold \(G/\Gamma\); \(G\)-invariant symplectic form; \(G\)-invariant harmonic form; symplectic harmonic cohomology group

Citations:

Zbl 0634.58029; Zbl 0872.58002; Zbl 1012.53076