For positive symplectic rational surfaces of type D, the group of Hamiltonian diffeomorphisms is a connected component of the symplectomorphism group in the C0-topology.
Topol (2002), 195--218
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$C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces
For positive symplectic rational surfaces of type D, the group of Hamiltonian diffeomorphisms is a connected component of the symplectomorphism group in the C0-topology.