Restrictions on symplectic fibrations (Q1877592)

The author studies symplectic bundles, investigating, in particular, how the additional structures (like Hamiltonicity) are reflected in the Leray-Serre spectral sequence of the bundle. One of main results is the following Theorem 1.5: Let \(\pi: P \to B\) be a Hamiltonian bundle with the (closed) symplectic fiber \((M,\omega)\) and \(P\) compact. Let \(i: M \to P\) denote the inclusion. Suppose that for any \(\alpha\in H^i(M), i\leq \dim M-4\) there exist \( \beta_0, \beta_1, \ldots \beta_k \in\operatorname{Im} i^*\) such that \[ \Phi_A(\alpha\cup\beta_0, \beta_1, \ldots , \beta_k)\neq 0 \] for some Gromov-Witten invariant \(\Phi_A\). Then the Leray-Serre spectral sequence of the bundle \(\pi\) collapses. This theorem is a partial affirmation of the well-known conjecture claiming that the Leray-Serre spectral sequence of a Hamiltonian bundle collapses. The author also uses the developed technique in order to estimate the rank of the flux group of a symplectic manifold (Theorem 1.4). This allows him to prove that the well-known flux conjecture holds for symplectic manifolds with hard Lefschetz property. The paper contains an Addendum entitled ``Simple examples of nontrivial Gromov-Witten invariants'' written jointly with Kaoru Ono.