  The ``Flux conjecture'' for symplectic manifolds states that the group of
Hamiltonian diffeomorphisms is C^1-closed in the group of all symplectic
diffeomorphisms. We prove the conjecture for spherically rational manifolds and
for those whose minimal Chern number on 2-spheres either vanishes or is large
enough. We also confirm a natural version of the Flux conjecture for symplectic
torus actions. In some cases we can go further and prove that the group of
Hamiltonian diffeomorphisms is C^0-closed in the identity component of the
group of all symplectic diffeomorphisms.
