On the Thom conjecture in $CP^3$

Abstract

What is the simplest smooth simply connected 4-manifold embedded in CP 3 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 manifoldThis contrasts with the Kronheimer-Mrowka solution of the Thom conjecture about surfaces in CP 2 , and is similar to results of Freedman for 2n-manifolds in CP n+1 with n odd and greater than 1.