  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 that, under certain assumptions, a Hamiltonian fibration c-splits, i.e.
the spectral sequence associated to the fibration collapses at E_2. Moreover,
we obtain a new estimate of the rank of flux groups.
  In the appendix, we provide examples of nonzero Gromov-Witten invariants for
blow-ups. As an application, we prove that any Hamiltonian fibration with fibre
being projective space CP^5 blown-up along 4-dimensional symplectic submanifold
c-splits.
