Topology Seminar: The Thom conjecture in CP^3
The homology groups of a smooth projective complex hypersurface in CP^n are determined by the degree of its defining equation. A question attributed to René Thom asks whether there is a smooth submanifold of CP^n with the same degree but with smaller Betti numbers. Kronheimer and Mrowka famously proved that the degree d curve in CP^2 has minimal Betti numbers, while Freedman showed many years ago that when n is even and at least4, the hypersurface does not minimize the Betti numbers. We show that this continues to hold when n = 3, finding smooth 4-manifolds embedded in CP^3 with degree at least 5 with b_2 smaller than the hypersurface of degree d.
This is joint work with Sašo Strle and Mark Slapar.
Monday, October 25 at 2:30pm
CW 122, CW 122