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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 $(M,\omega)$ be a compact symplectic $2n$-dimensional manifold, and let $F_\omega:\pi_1(\text{Symp}_0(M))\to H^1(M,\bbfR)$ be the flux homomorphism defined by $(F_\omega(\varphi),\alpha)= (\omega,\varphi_t\alpha)$, where $\text{Symp}_0(M)$ 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 $H^1(M,\bbfR)$ if and only if the Hamiltonian diffeomorphism group $\text{Ham}(M)$ is closed in $\text{Symp}_0 (M)$. The flux conjecture is the statement that $\Gamma_M$ is discrete. In this paper, first by using results of {\it F. Lalonde, D. McPuff} and {\it L. Polterovich} [On the flux conjectures, CRM Proc. Lect. Notes 15, 69--85 (1998; Zbl 0974.53062)], and {\it 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 $\text{Lef}_M:H^1(M, \bbfR)\to H^{2n-1}(M,\bbfR)$ is defined by $\text{Lef}_M(a)=a\cup \omega^{n-1}$. Then, applying this result to Donaldson submanifolds $(V^{2m},\omega)$, $1\le m\le n-1$, of $(M,\omega)$ [cf. {\it 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 {\it F. Lalonde, D. McPuff} and {\it 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 $(M,\omega)$ associated with $\varphi\in\pi_1(\text{Symp}_0 (M))$ are also studied (Cor. 1.8. proved in §4). For the entire collection see [Zbl 1089.53004].

53D35Global theory of symplectic and contact manifolds
57R17Symplectic and contact topology