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Flux conjecture on symplectic submanifolds. (English) Zbl 1099.53056
Mladenov, Ivaïlo (ed.) et al., Proceedings of the 7th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 2–10, 2005. Sofia: Bulgarian Academy of Sciences (ISBN 954-8495-30-9/pbk). 89-97 (2006).

Let $\left(M,\omega \right)$ be a compact symplectic $2n$-dimensional manifold, and let ${F}_{\omega }:{\pi }_{1}\left({\text{Symp}}_{0}\left(M\right)\right)\to {H}^{1}\left(M,ℝ\right)$ be the flux homomorphism defined by $\left({F}_{\omega }\left(\varphi \right),\alpha \right)=\left(\omega ,{\varphi }_{t}\alpha \right)$, where ${\text{Symp}}_{0}\left(M\right)$ is the identity component of the symplectomorphism group of $M$. Then, the flux group ${{\Gamma }}_{M}$ of $M$ is defined to be the image of ${F}_{\omega }$. It is a discrete subgroup of ${H}^{1}\left(M,ℝ\right)$ if and only if the Hamiltonian diffeomorphism group $\text{Ham}\left(M\right)$ is closed in ${\text{Symp}}_{0}\left(M\right)$. The flux conjecture is the statement that ${{\Gamma }}_{M}$ is discrete. In this paper, first by using results of F. Lalonde, D. McPuff and L. Polterovich [On the flux conjectures, CRM Proc. Lect. Notes 15, 69–85 (1998; Zbl 0974.53062)], and D. Gottlieb [Am. J. Math. 87, 840–856 (1965; Zbl 0148.17106)], ${{\Gamma }}_{M}$ is shown to be a subgroup of the kernel of the Lefschetz map ${\text{Lef}}_{M}$ if the Euler number of $M$ is not equal to zero (Th. 1.1. proved in §2). Here ${\text{Lef}}_{M}:{H}^{1}\left(M,ℝ\right)\to {H}^{2n-1}\left(M,ℝ\right)$ is defined by ${\text{Lef}}_{M}\left(a\right)=a\cup {\omega }^{n-1}$.

Then, applying this result to Donaldson submanifolds $\left({V}^{2m},\omega \right)$, $1\le m\le n-1$, of $\left(M,\omega \right)$ [cf. S. Donaldson, J. Differ. Geom. 44, No. 4, 666–705 (1996; Zbl 0883.53032)], and use authors previous results [Acta Math. Sin., Engl. Ser. 22, No. 1, 115–122 (2006; Zbl 1227.53090)] and results of F. Lalonde, D. McPuff and L. Porterovich [Invent. Math. 135, No.2, 369–385 (1999; Zbl 0907.58004)], flux conjecture is shown to hold for all Donaldson submanifolds of $M$ if the first Betti number of $M$ is equal to one (Th. 1.5. proved in §3). Applications of these results to the symplectic fibration over ${S}^{2}$ with fiber $\left(M,\omega \right)$ associated with $\varphi \in {\pi }_{1}\left({\text{Symp}}_{0}\left(M\right)\right)$ are also studied (Cor. 1.8. proved in §4).

##### MSC:
 53D35 Global theory of symplectic and contact manifolds 57R17 Symplectic and contact topology