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 be a compact symplectic -dimensional manifold, and let be the flux homomorphism defined by , where is the identity component of the symplectomorphism group of . Then, the flux group of is defined to be the image of . It is a discrete subgroup of if and only if the Hamiltonian diffeomorphism group is closed in . The flux conjecture is the statement that 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)], is shown to be a subgroup of the kernel of the Lefschetz map if the Euler number of is not equal to zero (Th. 1.1. proved in §2). Here is defined by .
Then, applying this result to Donaldson submanifolds , , of [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 if the first Betti number of is equal to one (Th. 1.5. proved in §3). Applications of these results to the symplectic fibration over with fiber associated with are also studied (Cor. 1.8. proved in §4).