The topology of a compact self-dual manifold whose twistor space has positive algebraic dimension is studied. When the algebraic dimension equals three, it is known by Campana  that the original self-dual manifold is homeomorphic to a connected sum of copies of a complex projecitve plane. In the remaining cases where the algebraic dimension is equal to two or one, we similarly determine the topology of the selfdual manifold except in a certain exceptional case where the algebraic dimension equals one.
"Topology of compact self-dual manifolds whose twistor space is of positive algebraic dimension." J. Math. Soc. Japan 54 (3) 587 - 608, July, 2002. https://doi.org/10.2969/jmsj/1191593910