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The GW/PT conjectures for toric pairs

Davesh Maulik, Dhruv Ranganathan

Abstract

We prove the conjectural correspondence between logarithmic Gromov-Witten theory and logarithmic Donaldson/Pandharipande-Thomas theory for pairs $(Y|\partial Y)$ consisting of a toric threefold $Y$ and any torus invariant divisor $\partial Y$, with primary insertions. The results are the first verifications of this conjecture when $\partial Y$ is singular, i.e. the ``fully logarithmic'' setting. When $\partial Y$ is empty, we get a new proof of the known toric correspondence, but our methods also lead to stronger conclusions. In particular, we show the PT series is a Laurent polynomial in the presence of sufficient positivity and prove a 2008 conjecture of Oblomkov, Okounkov, Pandharipande, and the first author stating the capped vertex is a Laurent polynomial. The methods also verify the logarithmic DT/PT conjecture for toric threefold pairs.

The GW/PT conjectures for toric pairs

Abstract

We prove the conjectural correspondence between logarithmic Gromov-Witten theory and logarithmic Donaldson/Pandharipande-Thomas theory for pairs consisting of a toric threefold and any torus invariant divisor , with primary insertions. The results are the first verifications of this conjecture when is singular, i.e. the ``fully logarithmic'' setting. When is empty, we get a new proof of the known toric correspondence, but our methods also lead to stronger conclusions. In particular, we show the PT series is a Laurent polynomial in the presence of sufficient positivity and prove a 2008 conjecture of Oblomkov, Okounkov, Pandharipande, and the first author stating the capped vertex is a Laurent polynomial. The methods also verify the logarithmic DT/PT conjecture for toric threefold pairs.
Paper Structure (57 sections, 36 theorems, 111 equations, 9 figures)

This paper contains 57 sections, 36 theorems, 111 equations, 9 figures.

Table of Contents

  1. Main results
  2. DT/PT
  3. Overview
  4. Broader context
  5. GW/PT
  6. Hodge integrals and triple DR cycles
  7. Refined tropical curve counting
  8. Logarithmic GW and PT theory
  9. Expansions
  10. Working up to subdivision
  11. Expansion from a polyhedral complex
  12. The target at a tropical point
  13. Stable maps and stable pairs
  14. GW evaluations
  15. PT evaluations
  16. ...and 42 more sections

Key Result

Theorem A

The logarithmic GW/PT correspondence holds equivariantly for toric threefold pairs.

Figures (9)

  • Figure 1: The map sending a $1$-complex to the corresponding configuration of points on the ends by taking asymptotics
  • Figure 2: A degeneration of $\mathbb P^2$ into elementary geometries, depicted by the polyhedral decomposition on the left, and the moment polytope complex on the right. On the right picture, the edges of each cell are toric boundary divisors. The edges that are contained within those of the outer triangle are non-boundary. Note that the quadratic singularities here have not been resolved.
  • Figure 3: A depiction of the ordering on stars induced by taking degenerations and then stars at vertices. In this case $\mathsf w$ on the right is smaller than $\mathsf v$.
  • Figure 4: The two Chow $1$-complexes, depending on the choice of constraint for the two vertical ends in the picture. The square indicates the end that carries the point condition, while the circles indicate divisorial conditions.
  • Figure 5: The $4$-valent star considered above.
  • ...and 4 more figures

Theorems & Definitions (87)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1.1.1
  • Definition 1.1.2
  • Remark 1.2.1: Non-uniqueness
  • Theorem 1.2.2
  • Remark 1.2.3
  • Remark 1.3.1
  • Conjecture 1.4.1: GW/PT correspondence
  • ...and 77 more