Restrictions on symplectic fibrations

Moduli spaces of stable pseudoholomorphic curves can be defined parametrically, i.e., over total spaces of symplectic fibrations. This imposes several restrictions on the spectral sequence of a symplectic fibration. We prove, among others, that under certain assumptions the spectral sequence collapses at E2. In the appendix, we prove nontriviality of certain Gromov–Witten invariant for blow-ups. As an application we obtain that any Hamiltonian fibration with the blow-up of CP^5 along four dimensional submanifold as a fibre c-splits. That is its spectral sequence collapses.