Virtual invariants of Quot schemes of points on threefolds

Abstract

We construct an almost perfect obstruction theory of virtual dimension zero on the Quot scheme parametrizing zero-dimensional quotients of a locally free sheaf on a smooth projective $3$-fold. This gives a virtual class in degree zero and therefore allows one to define virtual invariants of the Quot scheme. We compute these invariants proving a conjecture by Ricolfi. The computation is done by reducing to the toric case via cobordism theory and a degeneration argument. The toric case is solved by reducing to the computation on the Quot scheme of points on $\mathbb{A}^3$ via torus localization, the torus-equivariant Siebert's formula for almost perfect obstruction theories and the torus-equivariant Jouanolou trick.