Abstract
What is the simplest smooth simply connected 4-manifold embedded in CP3 homologous to a degree d hypersurface V-d? A version of this question associated with Thom asks if V-d has the smallest b(2) among all such manifolds. While this is true for degree at most 4, we show that for all d >= 5, there is a manifold M-d in this homology class with b(2)(M-d) < b(2) (V-d). This contrasts with the Kronheimer-Mrowka solution of the Thom conjecture about surfaces in CP2 and is similar to results of Freedman for 2n-manifolds in CPn+1 with n odd and greater than 1.