On the Feyzbakhsh-Thomas programme for Fano $3$-folds

Abstract

Let $X$ be a Fano $3$-fold with even canonical class which satisfies the generalized Bogomolov-Gieseker inequality, such as ${\mathbb{P}}^3$. We express Donaldson-Thomas invariants counting Gieseker semistable sheaves of rank $r$, where $r>0$, on $X$ in terms of those counting sheaves of rank $0$ and pure dimension $2$. This implements an analogue of the programme initiated by S. Feyzbakhsh and R. Thomas in the case of Calabi-Yau varieties. The methods include $K$-theoretic Donaldson-Thomas theory and, quite unexpectedly, the use of certain combinatorial properties of vertex algebras.

V.1, comments welcome