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Non-toric examples of elliptic canonical bases I

Tatsuyuki Hikita

Abstract

We propose several properties of elliptic lifts of the $K$-theoretic canonical bases for conical symplectic resolutions defined in our previous work. As an example, we construct elliptic lifts of canonical bases for the Hilbert scheme of 2-points in the affine plane.

Non-toric examples of elliptic canonical bases I

Abstract

We propose several properties of elliptic lifts of the -theoretic canonical bases for conical symplectic resolutions defined in our previous work. As an example, we construct elliptic lifts of canonical bases for the Hilbert scheme of 2-points in the affine plane.

Paper Structure

This paper contains 21 sections, 14 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Expected properties of elliptic canonical bases
  3. Dual pairs
  4. Elliptic bar involutions
  5. $K$-theory limits
  6. Duality of elliptic canonical bases
  7. Kähler $q$-difference equations
  8. Main conjecture
  9. $K$-theoretic canonical bases for Hilbert scheme of 2-points
  10. Elliptic stable bases
  11. $K$-theory limits
  12. $K$-theoretic canonical bases
  13. Elliptic canonical bases for Hilbert scheme of 2-points
  14. Main result
  15. Solving $q$-difference equations
  16. ...and 6 more sections

Key Result

Lemma 3.1

${\mathfrak{X}}$ is self dual under the identifications $a^!=z$, $z^!=a$, and $X^H\cong(X^!)^{H^!}$ given by $[2]^!=[1,1]$ and $[1,1]^!=[2]$.

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Conjecture 2.6
  • Conjecture 2.7
  • Conjecture 2.8
  • Lemma 3.1
  • Lemma 3.2
  • ...and 12 more