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Logarithmic negative tangency and root stacks

Luca Battistella, Navid Nabijou, Dhruv Ranganathan

Abstract

We study stable maps to normal crossings pairs with possibly negative tangency orders. There are two independent models: punctured Gromov-Witten theory of pairs and orbifold Gromov-Witten theory of root stacks with extremal ages. Exploiting the tropical structure of the punctured mapping space, we define and study a new virtual class for the punctured theory. This arises as a refined intersection product on the Artin fan, and produces a distinguished sector of punctured Gromov-Witten invariants. Restricting to genus zero, we show that these invariants coincide with the orbifold invariants, first for smooth pairs, and then for normal crossings pairs after passing to a sufficiently refined blowup. This builds on previous work to provide a complete picture of the logarithmic-orbifold comparison in genus zero, which is compatible with splitting and thus allows for the wholesale importation of orbifold techniques, including boundary recursion and torus localisation. Contemporaneous work of Johnston uses the comparison to give a new proof of the associativity of the Gross-Siebert intrinsic mirror ring.

Logarithmic negative tangency and root stacks

Abstract

We study stable maps to normal crossings pairs with possibly negative tangency orders. There are two independent models: punctured Gromov-Witten theory of pairs and orbifold Gromov-Witten theory of root stacks with extremal ages. Exploiting the tropical structure of the punctured mapping space, we define and study a new virtual class for the punctured theory. This arises as a refined intersection product on the Artin fan, and produces a distinguished sector of punctured Gromov-Witten invariants. Restricting to genus zero, we show that these invariants coincide with the orbifold invariants, first for smooth pairs, and then for normal crossings pairs after passing to a sufficiently refined blowup. This builds on previous work to provide a complete picture of the logarithmic-orbifold comparison in genus zero, which is compatible with splitting and thus allows for the wholesale importation of orbifold techniques, including boundary recursion and torus localisation. Contemporaneous work of Johnston uses the comparison to give a new proof of the associativity of the Gross-Siebert intrinsic mirror ring.
Paper Structure (41 sections, 19 theorems, 208 equations, 2 figures)

This paper contains 41 sections, 19 theorems, 208 equations, 2 figures.

Table of Contents

  1. Outline
  2. The key construction: the refined punctured theory
  3. Main results: punctured-orbifold comparisons
  4. Subtleties of negative tangency
  5. Proof highlights
  6. Birational invariance
  7. Prospects
  8. Refined punctured theory: construction and examples
  9. Numerical data for punctured maps
  10. Tropical punctured maps
  11. Puncturing substack
  12. Refined punctured theory
  13. Strata formulae for the refined virtual class and examples
  14. Puncturing rank one
  15. Regularly embedded puncturing substack
  16. ...and 26 more sections

Key Result

Theorem A

Let $(X|D)$ be a smooth pair. In genus zero, with fixed numerical data $\Lambda$, the refined punctured Gromov--Witten theory coincides with the orbifold Gromov--Witten theory up to a rooting factor. Precisely, fix $r$ sufficiently large and let $X_r$ denote the $r$th root stack of $X$ along $D$. Th Moreover $\upalpha$ induces an isomorphism on coarse spaces, while $\upomega$ is an isomorphism.

Figures (2)

  • Figure 1: The cone complex $\mathsf{T}$.
  • Figure 2: The cone complex $\mathsf{T}^\prime$.

Theorems & Definitions (61)

  • Remark 1: Global nature of the refined punctured theory
  • Theorem A: \ref{['thm: correspondence smooth pairs']}
  • Theorem B: \ref{['thm: main']}
  • Definition 1.1
  • Remark 1.2: Relationship to global contact orders
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • ...and 51 more