On the Thom conjecture in $CP^3$

TL;DR

This paper investigates the minimal second Betti number among smooth simply connected 4-manifolds embedded in complex projective 3-space, revealing that for degrees five and higher, manifolds with smaller b2 than hypersurfaces exist.

Contribution

It demonstrates that for degrees d ≥ 5, there are manifolds with smaller b2 than the degree d hypersurfaces, challenging the Thom conjecture in this context.

Findings

For d ≤ 4, hypersurfaces have minimal b2.

For d ≥ 5, manifolds with smaller b2 exist.

Contrasts with the Thom conjecture in CP^2.

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 $d \geq 5$, 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 $CP^2$, and is similar to results of Freedman for $2n$-manifolds in $CP^{n+1}$ with $n$ odd and greater than $1$.