Lectures on K-theoretic computations in enumerative geometry

  title={Lectures on K-theoretic computations in enumerative geometry},
  author={Andrei Okounkov},
  journal={arXiv: Algebraic Geometry},
  • A. Okounkov
  • Published 23 December 2015
  • Mathematics
  • arXiv: Algebraic Geometry
These are notes from my lectures on quantum K-theory of Nakajima quiver varieties and K-theoretic Donaldson-Thomas theory of threefolds given at Columbia and Park City Mathematics Institute. They contain an introduction to the subject and a number of new results. In particular, we prove the main conjecture of arXiv:hep-th/0412021 and the conjecture of arXiv:1404.2323 in the simplest case of reduced smooth curves. We also prove the the absence of quantum corrections to the capped vertex with… 
Quantum K-theory of flag varieties via non-abelian localization
We provide an explicit parameterization (a.k.a. ”reconstruction”) of the permutationinvariant big J -function of partial flag varieties, treated as a non-abelian GIT quotient of a linear space, by
A-type Quiver Varieties and ADHM Moduli Spaces
We study quantum geometry of Nakajima quiver varieties of two different types—framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection
Virtual Riemann-Roch Theorems for Almost Perfect Obstruction Theories
. 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
Equivariant K-theoretic enumerative invariants and wall-crossing formulae in abelian categories
We provide a general framework for wall-crossing of enumerative invariants in equivariant K-theory, by lifting the main homological constructions of [Joy21] in two directions: to K-theory, and to
K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface
Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of
K-theoretic enumerative geometry and the Hilbert scheme of points on a surface
K-theoretic enumerative geometry and the Hilbert scheme of points on a surface Noah Arbesfeld Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface
On the K-theory stable bases of the Springer resolution
Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in
Quantum K-Theory of Grassmannians and Non-Abelian Localization
In the example of complex grassmannians, we demonstrate various techniques available for computing genus-0 K-theoretic GW-invariants of flag manifolds and more general quiver varieties. In
ħ-Deformed Schubert Calculus in Equivariant Cohomology, K-Theory, and Elliptic Cohomology
In this survey paper we review recent advances in the calculus of Chern-Schwartz-MacPherson, motivic Chern, and elliptic classes of classical Schubert varieties. These three theories are
Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem
Motivic Chern classes are elements in the K-theory of an algebraic variety $X$ depending on an extra parameter $y$. They are determined by functoriality and a normalization property for smooth $X$.


Quantum Groups and Quantum Cohomology
In this paper, we study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, we construct a Hopf algebra Y_Q,
Etingof’s conjecture for quantized quiver varieties
We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing and provide an exact
On the WDVV equation in quantum K-theory.
0. Introduction. Quantum cohomology theory can be described in general words as intersection theory in spaces of holomorphic curves in a given Kahler or almost Kahler manifold X. By quantum K-theory
Quantum cohomology of the Hilbert scheme of points on A_n-resolutions
We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these
Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras
To Professor Shoshichi Kobayashi on his 60th birthday 1. Introduction. In this paper we shall introduce a new family of varieties, which we call quiver varieties, and study their geometric
Quantum difference equation for Nakajima varieties
For an arbitrary Nakajima quiver variety $X$, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the
Wreath Macdonald polynomials and categorical McKay correspondence
Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the
The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory
We completely characterize genus-0 K-theoretic Gromov-Witten invariants of a compact complex algebraic manifold in terms of cohomological Gromov-Witten invariants of this manifold. This is done by
Random surfaces enumerating algebraic curves
The discovery that a relation exists between the two topics in the title was made by physicists who viewed them as two approaches to Feynman integral over all surfaces in string theory: one via
In this article the author proves that the values of the multiplicative genera Ak under discussion, where K = 2, 3,..., are obstructions to the existence of nontrivial S1-actions on a unitary