Related papers: Virtual invariants of Quot schemes of points on th…
Isotropic Quot schemes parameterize rank $r$ isotropic subsheaves of a vector bundle equipped with symplectic or symmetric quadratic form. We define a virtual fundamental class for isotropic Quot schemes over smooth projective curves. Using…
We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K-theoretic invariants are…
Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and $K$-theoretic invariants for many moduli stacks of interest, including…
For a simple, rigid vector bundle $F$ on a Calabi-Yau $3$-fold $Y$, we construct a symmetric obstruction theory on the Quot scheme $\textrm{Quot}_Y(F,n)$, and we solve the associated enumerative theory. We discuss the case of other…
We show that the Quot scheme $\text{Quot}_{\mathbf{A}^3}(\mathcal{O}^r,n)$ admits a symmetric obstruction theory, and we compute its virtual Euler characteristic. We extend the calculation to locally free sheaves on smooth $3$-folds, thus…
We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological…
We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums…
For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by…
We construct virtual fundamental classes in all intersection theories including Chow theory, K-theory and algebraic cobordism for quasi-projective Deligne-Mumford stacks with perfect obstruction theories and prove the virtual pullback…
This is the third in a series of works devoted to constructing virtual structure sheaves and $K$-theoretic invariants in moduli theory. The central objects of study are almost perfect obstruction theories, introduced by Y.-H. Kiem and the…
Computing topological invariants of 3-manifolds is generally intractable, yet specialized algebraic structures can enable efficient algorithms. For Witten-Reshetikhin-Turaev (WRT) invariants of torus bundles, we exploit the non-commutative…
We study the virtual Euler characteristics of sheaves over Quot schemes of curves, establishing that these invariants fit into a topological quantum field theory (TQFT) valued in $\mathbb{Z}[[q]]$. We show that the three-pointed genus-zero…
We study punctual quot-schemes of torsion-free sheaves $E_Y$ on smooth projective curves, surfaces and Calabi--Yau fourfolds via their virtual geometry. Our goal is to give a complete description of the virtual fundamental classes and their…
We construct a virtual fundamental class on the Quot scheme parametrizing quotients of a trivial bundle on a curve. We use the virtual localization formula to calculate virtual intersection numbers on Quot. As a consequence, we reprove the…
Motivated by super-Yang-Mills theory on a Calabi-Yau 4-fold, Nekrasov and Piazzalunga have assigned weights to $r$-tuples of solid partitions and conjectured a formula for their weighted generating function. We define $K$-theoretic virtual…
We derive a $K$-theoretic analogue of the Vafa--Intriligator formula, computing the (virtual) Euler characteristics of vector bundles over the Quot scheme that compactifies the space of degree $d$ morphisms from a fixed projective curve to…
Let $\mathcal{E}$ be a locally free sheaf of rank $r$ on a smooth projective surface $S$. The Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ of length $l$ coherent sheaf quotients of $\mathcal{E}$ is a natural higher rank generalization…
We study Quot schemes of vector bundles on algebraic curves. Marian and Oprea gave a description of a topological quantum field theory (TQFT) studied by Witten in terms of intersection numbers on Quot schemes of trivial bundles. Since these…
We introduce the notion of almost perfect obstruction theory on a Deligne-Mumford stack and show that stacks with almost perfect obstruction theories have virtual structure sheaves which are deformation invariant. The main components in the…
Given an affine scheme X with an action of a reductive group G and a G-linearized coherent sheaf M, we construct the ``invariant Quot scheme'' that parametrizes the quotients of M whose space of global sections is a direct sum of simple…