# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\left(M,\omega \right)$ is a $2m$-dimensional symplectic manifold, one can define the space ${ℋ}_{\omega }^{k}\left(M\right)$ of all harmonic $k$-forms on $M$ and the symplectic harmonic $k$-th cohomology group ${H}_{\omega \text{-}hr}^{k}\left(M\right):={ℋ}_{\omega }^{k}\left(M\right)/{B}^{k}\left(M\right)\cap {ℋ}_{\omega }^{k}\left(M\right)$ [see J. Brylinski, J. Differ. Geom. 28, 93–114 (1988; 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 $𝔤$ 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 ${ℋ}_{\omega }^{k}\left(𝔤\right)$ be the space of all $G$-invariant harmonic $k$-forms on $G/{\Gamma }$ and ${H}_{\omega \text{-}hr}^{k}\left(𝔤\right):={ℋ}_{\omega }^{k}\left(𝔤\right)/{B}^{k}\left(𝔤\right)\cap {ℋ}_{\omega }^{k}\left(𝔤\right)$. The authors prove that ${B}^{3}\left(𝔤\right)\subset {ℋ}_{\omega }^{3}\left(𝔤\right)$ and some properties concerning $dim{H}_{\omega \text{-}hr}^{2m-2}\left(𝔤\right)$ and $dim{H}_{\omega \text{-}hr}^{2m-1}\left(𝔤\right)$. In particular, the dimension of ${H}_{\omega \text{-}hr}^{2m-1}\left(𝔤\right)$ 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 $G/{\Gamma }\phantom{\rule{4pt}{0ex}}\left(m\ge 6\right)$ admitting symplectic structures $\omega$ such that the dimension of ${H}_{\omega \text{-}hr}^{3}\left(𝔤\right)$ varies [see also D. Yan, Adv. Math. 120, 143–154 (1996; Zbl 0872.58002)], for a question raised by Khesin and McDuff].
##### MSC:
 53D35 Global theory of symplectic and contact manifolds 57R17 Symplectic and contact topology