Virtual invariants of Quot schemes of points on threefolds

In this thesis, we study the deformation theory and the enumerative geometry of the Quot scheme of points Quot_Y(F, n) parametrizing zero-dimensional quotients of length n of a locally free sheaf F on a smooth (quasi)-projective 3-fold Y. We construct on Quot_Y(F, n) an almost perfect obstruction theory of virtual dimension zero. The notion of an almost perfect obstruction theory recently introduced by Kiem--Savvas is a weaker notion of a perfect obstruction theory in the sense of Behrend--Fantechi which still gives a virtual class. We therefore obtain a Chow class [Quot_Y(F, n)]^vir in degree zero, which in the projective case allows one to define the virtual invariants by taking the degree. We compute the virtual invariants of Quot_Y(F, n) with Y projective proving a conjectural formula of Ricolfi. We first perform the computation in the toric case by reducing it to the one of Fasola–Monavari–Ricolfi. We have to reprove the torus localization formula and the Siebert formula for almost perfect obstruction theories under suitable assumptions. To this end, we exploit the Jouanolou trick. The computation in the general case follows from the toric case via the theory of double point cobordism of Lee–Pandharipande and a degeneration argument similar to the one in Li–Wu. In the course of the proof, we also introduce virtual invariants of the Quot scheme of points on a 3-fold relative to a smooth divisor D in Y. We compute these invariants and generalize results of Levine--Pandharipande to the higher rank case. The proof crucially uses the computation of the invariants for local surfaces. We also define the virtual invariants via localization for any smooth quasi-projective 3-fold Y, which is acted on by a torus T, and such that the locally free sheaf F has a T-equivariant structure, and the fixed locus Y^T is proper. We compute these invariants extending again the work of Levine–Pandharipande. The strategy is to reduce the computation to the case of local geometries using deformation to the normal cone. In summary, the thesis provides a complete solution of the virtual enumerative geometry of the Quot scheme of points of a locally free sheaf on a smooth quasi-projective 3-fold, generalizing the classical case of the Hilbert scheme of points on a 3-fold.