Positive 2-spheres in 4-manifolds of signature (1<i>,n</i>)

Abstract

We sharpen Donaldson's theorem on the standardness of definite intersection forms of smooth 4-manifolds in the same sense as Kervaire and Milnor sharpened Rohlin's signature theorem.We then apply the result thus obtained to show that the homology classes of rational surfaces with b^ < 9 which can be represented by smoothly embedded 2-spheres S with S S > 0 are up to diίfeomorphism represented by smooth rational curves.Furthermore, we not only extend part of the application to the case where b^ > 9, but also give an algorithm to see whether or not a given homology class of rational surfaces with b^ < 9 can be represented by a smoothly embedded 2-sphere.