Gromov-Witten theory of $\mathsf{Hilb}^n(\mathbb{C}^2)$ and Noether-Lefschetz theory of $\mathcal{A}_g$

TL;DR

This paper computes the genus 1 Gromov-Witten invariants of the Hilbert scheme of points in the plane, revealing connections to Noether-Lefschetz theory of abelian varieties and establishing a homomorphism property for cycle classes.

Contribution

It provides the first explicit calculation of genus 1 Gromov-Witten invariants for $ ext{Hilb}^n( ext{C}^2)$ and links these to Noether-Lefschetz geometry of $ ext{A}_g$, introducing new relations in the cycle classes.

Findings

Exact match between Gromov-Witten invariants and Noether-Lefschetz calculations.

Proved a homomorphism property for cycle classes in $ ext{CH}^*( ext{A}_g)$.

Full genus 1 Gromov-Witten theory determined modulo a conjecture.

Abstract

We calculate the genus 1 Gromov-Witten theory of the Hilbert scheme $\mathsf{Hilb}^n(\mathbb{C}^2)$ of points in the plane. The fundamental 1-point invariant (with a divisor insertion) is calculated using a correspondence with the families local curve Gromov-Witten theory over the moduli space $\overline{\mathcal{M}}_{1,1}$. The answer exactly matches a parallel calculation related to the Noether-Lefschetz geometry of the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties. As a consequence, we prove that the associated cycle classes satisfy a homomorphism property for the projection operator on $\mathsf{CH}^*(\mathcal{A}_g)$. The fundamental 1-point invariant determines the full genus 1 Gromov-Witten theory of $\mathsf{Hilb}^n(\mathbb{C}^2)$ modulo a nondegeneracy conjecture about the quantum cohomology. A table of calculations is given.