On optimal 4-dimensional metrics (Q941861)

The problem is studied, which simply connected oriented 4-manifolds admit anti-self dual Riemannian metrics of flat scalar curvature. The new key ingredient is a proof that the connected sum of five reverse-oriented projective planes \(\overline{\mathbb{C}\mathbb{P}_2}\) admits such metrics. It is proceeded on the basis of the main following theorem: A smooth compact simply connected 4-manifold admits the above said metrics if \(*\) \(M\) is diffeomorphic to \(k\,\overline{\mathbb{C}\mathbb{P}_2}\), for some \(k\geq 5\). \(*\) \(M\) is diffeomorphic to \(k\,\overline{\mathbb{C}\mathbb{P}_2}\# \overline{\mathbb{C}\mathbb{P}_2}\), for some \(k\geq 10\). \(*\) \(M\) is diffeomorphic to a hyper-Kähler manifold K3.