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On Hilb/Sym correspondence for K3 surfaces

Deniz Genlik, Hsian-Hua Tseng

TL;DR

The paper establishes a genus 0, 3-point reduced crepant-resolution correspondence between the Hilbert scheme of $n$ points on a K3 surface $S$ and the symmetric product $\mathsf{Sym}^n(S)$. It does so by adapting Nesterov's wall-crossing framework to the reduced setting, combining it with Gromov–Witten/Pairs correspondence results, and then computing explicit generating functions on the Hilbert side that match the Sym side under the appropriate change of variables. A concrete calculation for a Bryan–Leung K3 surface yields a modular-form–type expression, illustrating the structure of reduced invariants. The results extend the reach of CRC to reduced theories in the K3 context and highlight deep links between GW theory, relative/paired theories, and Hilbert-symmetric correspondences in genus 0.

Abstract

We derive a crepant resolution correspondence for some genus zero reduced Gromov-Witten invariants of Hilbert schemes of points on a K3 surface.

On Hilb/Sym correspondence for K3 surfaces

TL;DR

The paper establishes a genus 0, 3-point reduced crepant-resolution correspondence between the Hilbert scheme of points on a K3 surface and the symmetric product . It does so by adapting Nesterov's wall-crossing framework to the reduced setting, combining it with Gromov–Witten/Pairs correspondence results, and then computing explicit generating functions on the Hilbert side that match the Sym side under the appropriate change of variables. A concrete calculation for a Bryan–Leung K3 surface yields a modular-form–type expression, illustrating the structure of reduced invariants. The results extend the reach of CRC to reduced theories in the K3 context and highlight deep links between GW theory, relative/paired theories, and Hilbert-symmetric correspondences in genus 0.

Abstract

We derive a crepant resolution correspondence for some genus zero reduced Gromov-Witten invariants of Hilbert schemes of points on a K3 surface.
Paper Structure (17 sections, 2 theorems, 46 equations)

This paper contains 17 sections, 2 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. The geometry
  3. Reduced Gromov-Witten theory
  4. Crepant resolutions
  5. Calculations
  6. Outline
  7. Acknowledgment
  8. Gromov-Witten/Hurwitz wall-crossing
  9. Review of Nesterov's work
  10. Admissible maps
  11. $\epsilon=1$
  12. $\epsilon=0^+$
  13. Master space
  14. The reduced case
  15. Crepant resolution correspondence
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose $\gamma\in H_2(S,\mathbb{Z})$ is a nonzero class of divisibility at most $2$. Then the generating series $\langle \gamma_1,\gamma_2,\gamma_3\rangle_{0,\gamma}^{\mathsf{Hilb}^n(S), red}(y)$ is the Taylor expansion at $y=0$ of a rational function in $y$, and under the change of variables $-y=e

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Remark 3: G. Oberdieck
  • Remark 1