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Remarks on gluing punctured logarithmic maps

Mark Gross

TL;DR

This work develops a streamlined gluing formalism for punctured stable log maps in the Abramovich–Chen–Gross–Siebert program, enabling robust gluing formulas for log Gromov–Witten invariants across broad degenerations. Central to the approach are tropical tools: a tropical gluing map $\Psi$ and multiplicities $\mu(\boldsymbol{\tau},\mathbf{E})$, together with four-point lemmas that govern fs fibre products of log points and the gluing of moduli spaces. The framework recovers classical degeneration results (Li–Ruan, Jun Li) in a natural way and extends to settings with refined combinatorics, including cases where the gluing is tropically transverse. The paper then applies these methods to type III degenerations of K3 surfaces, deriving canonical wall structures via invariants for Looijenga pairs and establishing how blow-ups affect these counts, thereby contributing to mirror-symmetric descriptions of K3 moduli. Overall, the results unify and extend gluing in log Gromov–Witten theory and provide concrete tools for studying wall structures in mirror symmetry contexts.

Abstract

We consider some well-behaved cases of the gluing formalism for punctured stable log maps of Abramovich-Chen-Gross-Siebert. This gives a gluing formula for log Gromov-Witten invariants in a diverse set of cases; in particular, the gluing formulae of Li-Ruan, Jun Li and Kim-Lho-Ruddat become an easy special case. The last section gives an application of this gluing formalism to canonical wall structures for K3 surfaces as constructed by Gross and Siebert in "The canonical wall structure and intrinsic mirror symmetry."

Remarks on gluing punctured logarithmic maps

TL;DR

This work develops a streamlined gluing formalism for punctured stable log maps in the Abramovich–Chen–Gross–Siebert program, enabling robust gluing formulas for log Gromov–Witten invariants across broad degenerations. Central to the approach are tropical tools: a tropical gluing map and multiplicities , together with four-point lemmas that govern fs fibre products of log points and the gluing of moduli spaces. The framework recovers classical degeneration results (Li–Ruan, Jun Li) in a natural way and extends to settings with refined combinatorics, including cases where the gluing is tropically transverse. The paper then applies these methods to type III degenerations of K3 surfaces, deriving canonical wall structures via invariants for Looijenga pairs and establishing how blow-ups affect these counts, thereby contributing to mirror-symmetric descriptions of K3 moduli. Overall, the results unify and extend gluing in log Gromov–Witten theory and provide concrete tools for studying wall structures in mirror symmetry contexts.

Abstract

We consider some well-behaved cases of the gluing formalism for punctured stable log maps of Abramovich-Chen-Gross-Siebert. This gives a gluing formula for log Gromov-Witten invariants in a diverse set of cases; in particular, the gluing formulae of Li-Ruan, Jun Li and Kim-Lho-Ruddat become an easy special case. The last section gives an application of this gluing formalism to canonical wall structures for K3 surfaces as constructed by Gross and Siebert in "The canonical wall structure and intrinsic mirror symmetry."
Paper Structure (16 sections, 31 theorems, 146 equations, 1 figure)

This paper contains 16 sections, 31 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tropical maps and moduli of punctured curves
  4. The gluing formalism
  5. Four-point lemmas: fibre products of log points
  6. Gluing one curve
  7. Gluing moduli spaces
  8. The general situation
  9. Gluing from rigid tropical curves
  10. Punctured versus relative
  11. The classical degeneration formula
  12. Applications to wall structures for type III degenerations of K3 surfaces
  13. Review of wall types and balancing
  14. An invariant of Looijenga pairs
  15. Gluing via Parker and Ranganathan's triple point method
  16. ...and 1 more sections

Key Result

Theorem 1.1

If $W$ is non-empty, then it has $\mu({\boldsymbol{\tau}},\mathbf{E})$ connected components.

Figures (1)

  • Figure 1: The boundary divisor $\widetilde{\mathcal{D}}$ of $\widetilde{\mathcal{X}}$ and the corresponding affine manifold $\widetilde{B}'$. The vertices are all singular points of the affine structure.

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.4
  • Theorem 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Lemma 3.1
  • proof
  • ...and 60 more