arXiv subjects
Publications and source records attributed to Riccardo Ontani.
We express integrals over virtual cycles of GIT quotients $X/\!\!/G$ in terms of integrals over virtual cycles of fixed loci $X^T$. The results hold for both perfect obstruction theories and $(-2)$-shifted symplectic structures, in cohomology and in $K$-theory, and for noncompact Deligne-Mumford stacks acted on by a reductive group with compact quotient.
We study the virtual intersection theory of Hyperquot schemes parameterizing sequences of quotient sheaves of a vector bundle on a smooth projective curve. Our results generalize the Vafa--Intriligator formula for Quot schemes and provide a closed formula for virtual counts of maps from the curve to a partial flag variety.
We consider genus zero quasimap invariants of smooth projective targets of the form $V/\!/G$, where $V$ is a representation of a reductive group $G$. In particular we consider integrals of cohomology classes arising as characteristic classes of the universal quasimap. In this setting, we provide a way to express the invariants of $V/\!/G$ in terms of invariants of $V/\!/T$, where $T$ is a maximal subtorus of $G$. Using this, we obtain residue formulae for such invariants as conjectured by Kim, Oh, Yoshida and Ueda. Finally, under some positivity assumptions on $V/\!/G$, we prove a Vafa-Intriligator formula for the generating series of such invariants, expressing them as finite sums of explicit contributions.
We prove an equality, predicted in the physical literature, between the Jeffrey-Kirwan residues of certain explicit meromorphic forms attached to a quiver without loops or oriented cycles and its Donaldson-Thomas type invariants. In the special case of complete bipartite quivers we also show independently, using scattering diagrams and theta functions, that the same Jeffrey-Kirwan residues are determined by the the Gross-Hacking-Keel mirror family to a log Calabi-Yau surface.
We use an equivariant version of the localization formula of Jeffrey and Kirwan to prove a formula for virtual invariants $(\text{DT}$, $χ_y$, $\text{Ell})$ of critical loci in quotients of linear spaces by actions of reductive algebraic groups. In particular we recover formulae for the invariants of critical loci of potentials in moduli spaces of quiver representations predicted by physicists.
A common approach to the theory of nonlocal Poisson brackets, seen from the operatorial point of view, has been to keep implicit the sets on which these brackets act. In this paper we aim to explicitly define appropriate functional spaces underlying to the theory of 1 codimensional weakly nonlocal Poisson brackets, motivating the definitions, and to prove the validity in this context of some classical results in the field. We start by introducing the spaces for the local case, which will serve as building tools for those in the nonlocal one. The definition and the study of these nonlocal functionals are the core of this work; in particular we work out a characterization of the variational derivative of such objects. We then translate everything to the level of manifolds, defining a global version of the functionals, and introduce the notion nonlocal Poisson brackets in this context. We conclude by applying all the machinery to prove a theorem due to Ferapontov. This last application is the natural conclusion of our discussion and shows that the spaces we introduce are suitable objects to work with when studying topics in this theory.