Thom conjecture, that was originally asked in $\mathrm{CP}^2$, and is now proven for symplectic 4 manifolds, states that complex curves (symplectic surfaces) are genus minimizing in their homology class. (The $\mathrm{CP}^2$ case was proven by Kronheimer and Mrowka '94, the symplectic case by Ozsvath and Szabo '00.)    

Does anybody know if there is any analogous work being done in $\mathrm{CP}^3$? More concretely, is there any reason to think that complex hypersurfaces in $\mathrm{CP}^3$ are topologically the simplest oriented 4 manifolds in their homology classes. The simplest perhaps in a sense that the sum of betti numbers is minimal.