Smooth Structures and Einstein Metrics on $CP^2#5,6,7\bar{CP^2}$

We show that each of the topological 4-manifolds $CP^2#k\bar{CP^2}, for $k = 6, 7$admits a smooth structure which has an Einstein metric of scalar curvature$s > 0$, a smooth structure which has an Einstein metric with$s < 0$and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We show that there are infinitely many manifolds homeomorphic non-diffeomorphic to$CP^2#5\bar{CP^2}$ which do not admit an Einstein metric. We also exhibit new higher dimensional examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature. The main ingredients are recent constructions of exotic symplectic or complex manifolds with small topological numbers.

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