Self-Duality and Connected Sums of Complex Projective Planes

This chapter discusses a theorem giving a sufficient condition for a self-dual 4-manifold to be diffeomorphic to nCP². It briefly mentions some results on self-dual structures with symmetries nCP². Combining the work of Orlik and Raymond on manifolds with smooth T²-actions, the work of Poon on anti-self-dual Hermitian surfaces and LeBrun’s observations on the topology of self-dual 4-manifolds , the chapter presents the following observation: if a non-negative type self-dual 4-manifold has torus symmetry with fixed points then the manifold is homeomorphic to nCP².