On the Thom conjecture in $C{P}^3$

Abstract

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

13 pages, 2 figures

References in corpus (1)

• On the topological 4-genus of torus knots