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Birational Invariance in Punctured Log Gromov-Witten Theory

Samuel Johnston

TL;DR

This work extends birational invariance results from classical log Gromov–Witten theory to punctured log Gromov–Witten theory under log étale modifications. It introduces a universal modification stack \\mathfrak{M}_{\\gamma ightarrow au} and establishes cartesian diagrams and étaleness properties that align obstruction theories and virtual classes across targets X and \\tilde{X}, enabling comparison of punctured invariants. The paper further shows the invariance of canonical wall structures and intrinsic mirror algebras under such modifications, strengthening the connection between punctured GW theory and mirror symmetry. Decorated tropical types are developed to control ambiguities in lifts, boundedness results are obtained without global generation assumptions, and the results yield robust birational invariance for both canonical walls and mirror constructions in Gross–Siebert and intrinsic mirror frameworks.

Abstract

Given a log smooth scheme $(X,D)$, and a log étale modification $(\tilde{X},\tilde{D}) \rightarrow (X,D)$, we relate the punctured Gromov-Witten theory of $(\tilde{X},\tilde{D})$ to the punctured Gromov-Witten theory of $(X,D)$, generalizing results of Abramovich and Wise in the non-punctured setting in "Birational invariance in log Gromov-Witten Theory". Using the main comparison results, we show a form of log étale invariance for the logarithmic mirror algebras and canonical scattering diagrams constructed in "Intrinsic Mirror Symmetry" and "The Canonical Wall Structure and Intrinsic Mirror Symmetry" respectively.

Birational Invariance in Punctured Log Gromov-Witten Theory

TL;DR

This work extends birational invariance results from classical log Gromov–Witten theory to punctured log Gromov–Witten theory under log étale modifications. It introduces a universal modification stack \\mathfrak{M}_{\\gamma ightarrow au} and establishes cartesian diagrams and étaleness properties that align obstruction theories and virtual classes across targets X and \\tilde{X}, enabling comparison of punctured invariants. The paper further shows the invariance of canonical wall structures and intrinsic mirror algebras under such modifications, strengthening the connection between punctured GW theory and mirror symmetry. Decorated tropical types are developed to control ambiguities in lifts, boundedness results are obtained without global generation assumptions, and the results yield robust birational invariance for both canonical walls and mirror constructions in Gross–Siebert and intrinsic mirror frameworks.

Abstract

Given a log smooth scheme , and a log étale modification , we relate the punctured Gromov-Witten theory of to the punctured Gromov-Witten theory of , generalizing results of Abramovich and Wise in the non-punctured setting in "Birational invariance in log Gromov-Witten Theory". Using the main comparison results, we show a form of log étale invariance for the logarithmic mirror algebras and canonical scattering diagrams constructed in "Intrinsic Mirror Symmetry" and "The Canonical Wall Structure and Intrinsic Mirror Symmetry" respectively.
Paper Structure (11 sections, 34 theorems, 36 equations, 8 figures)

This paper contains 11 sections, 34 theorems, 36 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Logarithmic and Tropical preliminaries
  4. Tropical lifts and stabilization maps
  5. Universal modification $\mathfrak{M}_{\gamma \rightarrow \tau} \rightarrow \mathfrak{M}_\tau$
  6. Construction of the morphism $\mathfrak{M}_{\gamma \rightarrow \tau}\rightarrow \mathfrak{M}_\gamma$
  7. Proof of Main Theorems
  8. Decorated version of main theorem
  9. Boundedness of punctured log maps
  10. Birational invariance of canonical wall structures
  11. Birational invariance of intrinsic mirrors

Key Result

Theorem 1.1

Assume $\tau$ and $\gamma$ are a pair of realizable tropical types on $X$ and $\tilde{X}$, with $\gamma$ a lift of $\tau$, such that stabilization induces a morphism $s: \mathscr{M}(\tilde{X}/B,\gamma) \rightarrow \mathscr{M}(X/B,\tau)$. Then there exists a log algebraic stack $\mathfrak{M}_{\gamma

Figures (8)

  • Figure 3.1: An example of a puncturing morphism $\Gamma_{\sigma}^\circ \rightarrow \Gamma_{\sigma}$ restricted to a cone $\sigma_{l}$ associated with a leg. In this case, we have $\sigma \cong \mathbb{R}_{\ge 0}^2$, and the leg $l$ of one of the paramterized tropical curves is depicted. Note that unlike for edges, the length of the leg need not be a linear function on the base.
  • Figure 4.1: A graphical depiction of choices of tropical lift. The left hand graphic is a tropical map to $\Sigma(X)$ with associated tropical type $\tau$, and the right hand graphic depicts tropical maps to a subdivision $\Sigma(\tilde{X})$ of $\Sigma(X)$ whose tropical types are various choices of tropical lift of $\tau$. These choices includes various constraints on the tropical modulus, as well as a choice of leg length.
  • Figure 4.2: Different choices of puncturing of a leg of $\tau$ mapping to the depicted cone giving different tropical types. The leftmost depicted leg is maximally extended.
  • Figure 4.3: An example of how a subdivision of the tropical target induces a subdivision of the tropical moduli problem. The subdivision of the target above is a barycentric subdivision of $\mathbb{R}_{\ge 0}^3$, and the tropical type $\tau$ has a associated cone $\mathbb{R}_{\ge 0}^2$ given below.
  • Figure 4.4: An example showing the necessity of lattice coarsening to ensure integer valued edge lengths after a subdivision. Non-filled circles indicate integral points in $\tau_\mathbb{N} \setminus \gamma_{\mathbb{N}}$, after identifying $\gamma$ as a subcone of $\tau$.
  • ...and 3 more figures

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 60 more