On the Thom Conjecture in ℂℙ3

Abstract

What is the simplest smooth simply connected $4$-manifold embedded in $\mathbb {C}\mathbb {P}^3$ 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 $d \geq 5$, 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 $\mathbb {C}\mathbb {P}^2$ and is similar to results of Freedman for $2n$-manifolds in $\mathbb {C}\mathbb {P}^{n+1}$ with $n$ odd and greater than $1$.