Title: Application of a Bogomolov-Gieseker type inequality to counting invariants
Abstract: In the preliminary talk, I will first explain the notion of (weak) Bridgeland stability conditions on the bounded derived category of coherent sheaves on a smooth projective threefold. Then I will discuss the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda.
In the main talk: I will work on a smooth projective threefold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the projective space P^3 or the quintic threefold. I will show certain moduli spaces of 2-dimensional torsion sheaves on X are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in X. When X is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. In the end, I will sketch how we can generalise this method to higher ranks to express DT invariants counting Gieseker semistable sheaves of any rank > 1 on X in terms of those counting sheaves of rank 0 and pure dimension 2. This is joint work with Richard Thomas.