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Refined invariants for Abelian surfaces: between polynomiality and modularity

Thomas Blomme, Gurvan Mével

TL;DR

The paper studies tropical refined invariants for abelian surfaces and connects them to reduced Gromov–Witten theory, extending polynomiality phenomena from the toric setting. The central result shows that for fixed genus $g$, the asymptotic refined invariant $AR_g^\star(n,x)$ is a polynomial in $n$ whose $x$-dependent coefficients are quasi-modular forms, namely $AR_g^\star(n,x)= ng-1+\sum_{k=1}^{g-3} f_k(x)n^k$ with $f_k$ vanishing at $0$. A key technical device is the introduction of the auxiliary family of quasi-modular forms $G_m(x)$, which arise in the combinatorial analysis of the sums and enable a clean expression in terms of Eisenstein series and derivatives. The authors also compute explicit low-genus examples (e.g., genus 2–4) and the first coefficients in fixed codegree, illustrating the interplay between Laurent-polynomial data and genuine polynomials, and supporting the conjectured modular structure of coefficients. Overall, the work strengthens the link between tropical counts, modular forms, and Gromov–Witten theory in the abelian setting, aligning with broader conjectures about bounded degree and quasi-modularity of coefficients.

Abstract

Tropical refined invariants for toric surfaces, introduced Block and G{ö}ttsche, are obtained couting tropical curves with a Laurent polynomial multiplicity. Brugall{é} and Jaramillo-Puentes then exhibited a polynomial behavior of the coefficients of this Laurent polynomial, seen as function on the curve degree. The authors provided explicit formula for small genus, involving quasi-modular forms. Inspired by the toric setting, the first-named author defined refined invariants for abelian surfaces and extended the polynomiality result. In this paper, we further study this regularity for abelian surfaces, providing explicit formulas involving quasi-modular forms. This resonates with the small genus cases of the toric setting.

Refined invariants for Abelian surfaces: between polynomiality and modularity

TL;DR

The paper studies tropical refined invariants for abelian surfaces and connects them to reduced Gromov–Witten theory, extending polynomiality phenomena from the toric setting. The central result shows that for fixed genus , the asymptotic refined invariant is a polynomial in whose -dependent coefficients are quasi-modular forms, namely with vanishing at . A key technical device is the introduction of the auxiliary family of quasi-modular forms , which arise in the combinatorial analysis of the sums and enable a clean expression in terms of Eisenstein series and derivatives. The authors also compute explicit low-genus examples (e.g., genus 2–4) and the first coefficients in fixed codegree, illustrating the interplay between Laurent-polynomial data and genuine polynomials, and supporting the conjectured modular structure of coefficients. Overall, the work strengthens the link between tropical counts, modular forms, and Gromov–Witten theory in the abelian setting, aligning with broader conjectures about bounded degree and quasi-modularity of coefficients.

Abstract

Tropical refined invariants for toric surfaces, introduced Block and G{ö}ttsche, are obtained couting tropical curves with a Laurent polynomial multiplicity. Brugall{é} and Jaramillo-Puentes then exhibited a polynomial behavior of the coefficients of this Laurent polynomial, seen as function on the curve degree. The authors provided explicit formula for small genus, involving quasi-modular forms. Inspired by the toric setting, the first-named author defined refined invariants for abelian surfaces and extended the polynomiality result. In this paper, we further study this regularity for abelian surfaces, providing explicit formulas involving quasi-modular forms. This resonates with the small genus cases of the toric setting.
Paper Structure (26 sections, 9 theorems, 54 equations)

This paper contains 26 sections, 9 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Toric case
  4. Abelian case
  5. Results and organisation of the paper
  6. A summary on refined invariants for abelian surfaces
  7. Tropical refined invariants for Abelian surfaces
  8. Tropical tori and tropical curves
  9. Enumerative problem and invariants
  10. Explicit computation
  11. Curves in a fixed linear system
  12. Connection to reduced Gromov-Witten invariants
  13. Reduced Gromov-Witten invariants
  14. Generating series of invariants with a $\lambda$-class
  15. Explicit computation
  16. ...and 11 more sections

Key Result

Proposition 2.3

Let $\mathbb{T} A$ be a tropical torus given by $S\colon\Lambda\hookrightarrow\mathbb{R}^2$. Then $B\colon\Lambda^*\to\mathbb{Z}^2$ is a polarization if and only if $BS^\intercal\colon(\mathbb{R}^2)^*\to\mathbb{R}^2$ induces a symmetric and positive definite bilinear form.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4: blomme2022abelian3
  • Remark 2.5
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 19 more