Gromov-Witten theory of Hilbⁿ(ℂ²) and Noether-Lefschetz theory of A_g

2025/06/14 by Lopez, Aitor Iribar, Pandharipande, Rahul, Tseng, Hsian-Hua
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.12438

Abstract

We calculate the genus 1 Gromov-Witten theory of the Hilbert scheme Hilbⁿ(ℂ²) 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 M_1,1. The answer exactly matches a parallel calculation related to the Noether-Lefschetz geometry of the moduli space 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 CH^*(A_g). The fundamental 1-point invariant determines the full genus 1 Gromov-Witten theory of Hilbⁿ(ℂ²) modulo a nondegeneracy conjecture about the quantum cohomology. A table of calculations is given.

Citations

• Noether-Lefschetz cycles on the moduli space of abelian varieties
• Tautological and non-tautological cycles on the moduli space of abelian varieties
• d-elliptic loci and the Torelli map
• Tautological projection for cycles on the moduli space of abelian varieties
• Quantum cohomology of the Hilbert scheme of points on an elliptic surface
• Hilbert schemes of points on surfaces and multiple q-zeta values
• The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves
• Cohomological field theory calculations
• The quantum differential equation of the Hilbert scheme of points in the\n plane
• Exts and vertex operators
• The Crepant Resolution Conjecture
• The local Donaldson-Thomas theory of curves
• Quantum cohomology of the Hilbert scheme of points in the plane
• The local Gromov-Witten theory of curves
• Gromov-Witten theory and Donaldson-Thomas theory, II
• Gromov-Witten theory and Donaldson-Thomas theory, I
• Gromov-Witten theory, Hurwitz theory, and completed cycles
• Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
• Semisimple Frobenius structures at higher genus
• Hodge integrals and Gromov-Witten theory
• Elliptic Gromov - Witten invariants and the generalized mirror\n conjecture
• Cycles on the Moduli Space of Abelian Varieties
• Heisenberg algebra and Hilbert schemes of points on projective surfaces
• Instantons and affine algebras I: The Hilbert scheme and vertex operators
• Algebraic Families on an Algebraic Surface

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• The local Gromov-Witten theory of curves