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Punctured logarithmic maps

Dan Abramovich, Qile Chen, Mark Gross, Bernd Siebert

Abstract

We introduce a variant of stable logarithmic maps, which we call punctured logarithmic maps. They allow an extension of logarithmic Gromov-Witten theory in which marked points have a negative order of tangency with boundary divisors. As a main application we develop a gluing formalism which reconstructs stable logarithmic maps and their virtual cycles without expansions of the target, with tropical geometry providing the underlying combinatorics. Punctured Gromov-Witten invariants also play a pivotal role in the intrinsic construction of mirror partners by the last two authors in arXiv:1909.07649, conjecturally relating to symplectic cohomology, and in the logarithmic gauged linear sigma model in upcoming work of the second author with Felix Janda and Yongbin Ruan.

Punctured logarithmic maps

Abstract

We introduce a variant of stable logarithmic maps, which we call punctured logarithmic maps. They allow an extension of logarithmic Gromov-Witten theory in which marked points have a negative order of tangency with boundary divisors. As a main application we develop a gluing formalism which reconstructs stable logarithmic maps and their virtual cycles without expansions of the target, with tropical geometry providing the underlying combinatorics. Punctured Gromov-Witten invariants also play a pivotal role in the intrinsic construction of mirror partners by the last two authors in arXiv:1909.07649, conjecturally relating to symplectic cohomology, and in the logarithmic gauged linear sigma model in upcoming work of the second author with Felix Janda and Yongbin Ruan.

Paper Structure

This paper contains 92 sections, 82 theorems, 324 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Scope and motivation
  3. Main features of punctured Gromov-Witten theory
  4. Statements of main results
  5. Organization of the paper
  6. Relation to other work
  7. Acknowledgements
  8. Conventions
  9. Punctured maps
  10. Definitions
  11. Puncturing
  12. Pre-stable punctured log structures
  13. Pull-backs of puncturings
  14. Punctured curves
  15. Pull-backs of punctured curves
  16. ...and 77 more sections

Key Result

Proposition 2.5

Let $X$ be a fine log scheme, and $Y$ as in Definition def:puncturing, with a choice of puncturing $Y^{\circ}$ and a morphism $f:Y^{\circ} \rightarrow X$. Let $\widetilde{Y}^{\circ}$ denote the puncturing of $Y$ given by the subsheaf of $\mathcal{M}_{Y^{\circ}}$ generated by $\mathcal{M}_Y$ and $f^{

Figures (13)

  • Figure 1: A puncturing $Y^\circ$ of a monoid $\mathcal{M}=\mathcal{M}_W$. Note that the part with negative projection in $\mathcal{P}^{{\mathrm{gp}}}$ (open circles) is not necessarily saturated.
  • Figure 2: A morphism of the previous puncturing $Y^\circ$ which is not pre-stable, with $f^\flat \mathcal{M}_X$ generated by $(2,-1)$. The submonoid generated by $\mathcal{M}_Y$ and $f^\flat \mathcal{M}_X$, shown in solid dots, is a different puncturing $\widetilde{Y}^{\circ}$ which is pre-stable.
  • Figure 3: The solid puncturing on the left extends to $\Bbbk[\epsilon]/(\epsilon^2)$ but no further --- the circled elements are the ones allowed for $k=1$. Its pull-back (see below) via $\mathcal{E}^2 = \epsilon$ is pictured on the right --- it is defined on $\Bbbk[\mathcal{E}]/(\mathcal{E}^4)$ but does not extend further.
  • Figure 4: The $(-1)$-curve and its monoid.
  • Figure 5: The length of a bounded leg varies piecewise linearly under linear variations of the adjacent vertex. The figure shows the intersection of the situation with an affine hyperplane.
  • ...and 8 more figures

Theorems & Definitions (229)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • Definition 2.6
  • Corollary 2.7
  • Proposition 2.8
  • proof
  • Definition 2.9
  • ...and 219 more