Diffeomorphisms, symplectic forms and Kodaira fibrations

As was recently pointed out by McMullen and Taubes [7], there are 4-manifolds for which the diffeomorphism group does not act transitively on the deformation classes of orientation-compatible symplectic structures. This note points out some other 4-manifolds with this property which arise as the orientation-reversed versions of certain complex surfaces constructed by Kodaira [3]. While this construction is arguably simpler than that of McMullen and Taubes, its simplicity comes at a price: the examples exhibited herein all have large fundamental groups.