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Gromov-Witten Invariants of Bielliptic Surfaces

Thomas Blomme

TL;DR

This work develops a concrete floor/pearl-diagram framework to compute Gromov-Witten invariants of bielliptic surfaces, leveraging Li degeneration and a decomposition into relative invariants of $ ilde E imesP^1$ to enumerate curves via pearl diagrams. It proves that generating series for enumerative GW-invariants are quasi-modular, with separate modular behavior in the elliptic directions, governed by $SL_2(Z)$ and congruence subgroups Γ_1(n). The paper further refines these invariants by inserting λ-classes, derives refined multiplicities for diagrams, and demonstrates how these refinements preserve modularity properties through multi-variable quasi-Jacobi form theory. Explicit genus-by-genus computations illustrate the structure, including the appearance of Eisenstein series and their twists, and the results point to broad future avenues in descendant and torsion-related refinements. Overall, the approach provides a rigorous, modularity-driven toolkit for exact GW-counts on bielliptic surfaces with potential extensions to other elliptic fibrations and refined enumerative theories.

Abstract

Bielliptic surfaces appear as quotient of a product of two elliptic curves and were classified by Bagnera-Franchis. We give a concrete way of computing their GW-invariants with point insertions using a floor diagram algorithm. Using the latter, we are able to prove the quasi-modularity of their generating series by relating them to generating series of graphs for which we also prove quasi-modularity results. We propose a refinement of these invariants by inserting a λ-class in the considered GW-invariants.

Gromov-Witten Invariants of Bielliptic Surfaces

TL;DR

This work develops a concrete floor/pearl-diagram framework to compute Gromov-Witten invariants of bielliptic surfaces, leveraging Li degeneration and a decomposition into relative invariants of to enumerate curves via pearl diagrams. It proves that generating series for enumerative GW-invariants are quasi-modular, with separate modular behavior in the elliptic directions, governed by and congruence subgroups Γ_1(n). The paper further refines these invariants by inserting λ-classes, derives refined multiplicities for diagrams, and demonstrates how these refinements preserve modularity properties through multi-variable quasi-Jacobi form theory. Explicit genus-by-genus computations illustrate the structure, including the appearance of Eisenstein series and their twists, and the results point to broad future avenues in descendant and torsion-related refinements. Overall, the approach provides a rigorous, modularity-driven toolkit for exact GW-counts on bielliptic surfaces with potential extensions to other elliptic fibrations and refined enumerative theories.

Abstract

Bielliptic surfaces appear as quotient of a product of two elliptic curves and were classified by Bagnera-Franchis. We give a concrete way of computing their GW-invariants with point insertions using a floor diagram algorithm. Using the latter, we are able to prove the quasi-modularity of their generating series by relating them to generating series of graphs for which we also prove quasi-modularity results. We propose a refinement of these invariants by inserting a λ-class in the considered GW-invariants.
Paper Structure (64 sections, 29 theorems, 120 equations, 8 figures)

This paper contains 64 sections, 29 theorems, 120 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Bielliptic surfaces
  4. Enumerative problems and GW-invariants.
  5. Refinement of the enumerative invariants.
  6. Degeneration and enumeration techniques.
  7. Results
  8. Pearl diagram algorithm.
  9. Regularity of the generating series.
  10. Possible directions
  11. More GW-invariants.
  12. More precise computation and regularity.
  13. Regularity of the refined invariants.
  14. Extension of techniques to different situations.
  15. Organization of the paper
  16. ...and 49 more sections

Key Result

Theorem 1

(Proposition prop-decomposition and prop-computation-multiplicity) The count of pearl diagrams with multiplicity $m(\mathscr{P})$ yields the GW-invariant $\langle \mathrm{pt}^{g-1} \rangle_{g,\varpi}^S$.

Figures (8)

  • Figure 1: Summary of the classification of the bielliptic surface. On the second column, the parameter of the elliptic curve $E=\mathbb{C}/\langle 1,\tau \rangle$, third row the torsion part in the homology groups, and last, a basis of the second homology group modulo torsion.
  • Figure 2: Example of pearl diagram, map to $\mathbb{R}/\mathbb{Z}$ and associated discerning pearl diagram. Non-flat vertices are depicted as square-box, circle-boxes for flat-vertices, and dotted circles for non-preferred vertices.
  • Figure 3: Another example of pearl diagram, map to $\mathbb{R}/\mathbb{Z}$ and associated discerning pearl diagram.
  • Figure 4: Yet another example of pearl diagram, map to $\mathbb{R}/\mathbb{Z}$ and associated discerning pearl diagram.
  • Figure 5: A graph with a specified orientation and height define a map to $\mathbb{R}/\mathbb{Z}$.
  • ...and 3 more figures

Theorems & Definitions (100)

  • Theorem
  • Theorem
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 90 more