Joyce twistor space and the associated Kähler class

Fix a smooth action of a real two-torus on the connected sum of $m$ copies of complex projective plane. Then the invariant self-dual structures constructed by Joyce, and hence the associated twistor spaces, depend on real $(m-1)$-dimensional parameter. We can then associate each twistor space a Kaehler class on a fixed open rational surface, which makes the moduli space of Joyce twistor spaces a domain of the projectified Kähler cone of the surface. This result then gives a nice description of the families of anti-self-dual bihermitian structures on hyperbolic Inoue surfaces constructed previously with Pontecorvo.