arxiv: 2603.07772 · v2 · submitted 2026-03-08 · 🧮 math.AG

Recognition: 2 theorem links

· Lean Theorem

The GW/PT conjectures for toric pairs

Davesh Maulik , Dhruv Ranganathan

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Pith reviewed 2026-05-15 14:34 UTC · model grok-4.3

classification 🧮 math.AG

keywords logarithmic Gromov-Witten theorylogarithmic Donaldson-Thomas theoryPandharipande-Thomas theorytoric threefoldstorus-invariant divisorsenumerative geometryvirtual classes

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The pith

Logarithmic Gromov-Witten and Pandharipande-Thomas theories agree for toric threefold pairs with any invariant divisor.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that logarithmic Gromov-Witten and logarithmic Donaldson/Pandharipande-Thomas theories produce matching counts for any toric threefold equipped with a torus-invariant divisor, using primary insertions. This establishes the first results in the fully logarithmic setting where the divisor may be singular and gives the first equivariant proof when the divisor is nonempty. The argument shows that the constraints of the logarithmic theory, together with toric equivariance and positivity, reduce every such invariant to the single known degree-one series of projective space. A sympathetic reader cares because the result unifies two central enumerative theories and supplies explicit evaluations for a large class of varieties.

Core claim

We prove the conjectural correspondence between logarithmic Gromov-Witten theory and logarithmic Donaldson/Pandharipande-Thomas theory for pairs (Y|∂Y) consisting of a toric threefold Y and any torus invariant divisor ∂Y, with primary insertions. The results are the first verifications when ∂Y is singular and the first proof of the equivariant toric correspondence for pairs when ∂Y is nonempty. When ∂Y is empty the methods recover the known toric correspondence while also showing that the PT series is a Laurent polynomial under sufficient positivity and proving the 2008 conjecture that the capped vertex is a Laurent polynomial. The logarithmic DT/PT conjecture is verified for toric threefold

What carries the argument

The reduction of all logarithmic invariants of toric pairs to the degree-one series of projective space, achieved by combining the constraints of the logarithmic virtual class with toric equivariance and positivity.

If this is right

The Pandharipande-Thomas series is a Laurent polynomial whenever sufficient positivity holds.
The capped vertex is a Laurent polynomial, confirming the 2008 conjecture.
The logarithmic DT/PT correspondence holds for all toric threefold pairs.
Every invariant for such pairs is determined by one explicit calculation on projective space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same reduction strategy may apply to non-toric threefolds if comparable logarithmic constraints can be established.
Positivity conditions appear to control the algebraic form of the generating functions across these theories.
The unification supplies a route to recursive computation of higher-genus or descendant logarithmic invariants.

Load-bearing premise

The constraints of the logarithmic theory suffice to determine the complete evaluation of toric pairs from the single degree 1 series of P^3, relying on toric equivariance and positivity.

What would settle it

Direct computation of the logarithmic GW and PT series for a toric pair with singular boundary divisor in degree two or higher, which would disagree if the claimed equality fails.

read the original 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, and the first proof of the equivariant toric correspondence for pairs when $\partial Y$ is nonempty. 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. Using the constraints of the logarithmic theory, the complete evaluation of toric pairs is determined by a single calculation -- the degree $1$ series of $\mathbb{P}^3$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves the log GW/PT correspondence for toric pairs with singular divisors by reducing all primary invariants to the single degree-1 series on P^3.

read the letter

The main result is a proof of the logarithmic GW/PT correspondence for toric threefold pairs (Y|∂Y) with any torus-invariant divisor, including singular ones. This is the first such verification in the fully logarithmic case and the first equivariant toric proof with nonempty divisors. When the divisor is empty they recover the known toric case but with stronger conclusions, including that the PT series is a Laurent polynomial under sufficient positivity and that the capped vertex is a Laurent polynomial, confirming a 2008 conjecture of Oblomkov-Okounkov-Pandharipande-Maulik. They also verify the logarithmic DT/PT conjecture for these pairs. The key move is showing that the constraints of the logarithmic theory plus toric equivariance and positivity fix every primary invariant from just the degree-1 series on P^3. That reduction is the part that actually delivers the computational bridge. The approach looks clean for the smooth cases where toric localization already works, and the abstract indicates they extend the same logic to singular divisors by controlling the extra strata through the log structure. The citation pattern is standard and appropriate, with no obvious circularity since the base case is external. One place that could use extra scrutiny is whether the log constraints really eliminate all new independent classes that might appear in the obstruction theories for singular ∂Y; the paper asserts they do, but the details of how the additional strata are related back to the P^3 calculation would be the natural point for a referee to check. This is for people working on enumerative invariants, log geometry, or toric mirror symmetry who want explicit relations between GW and PT theories. It is worth a serious referee because the result is substantial, the methods are grounded in existing techniques, and the reduction gives something concrete that others can use.

Referee Report

2 major / 1 minor

Summary. The paper proves the conjectural correspondence between logarithmic Gromov-Witten theory and logarithmic Donaldson/Pandharipande-Thomas theory for pairs (Y|∂Y) consisting of a toric threefold Y and any torus-invariant divisor ∂Y, with primary insertions. It establishes this for singular ∂Y (the fully logarithmic setting) and provides the first equivariant toric proof when ∂Y is nonempty. All toric pair invariants are reduced to the single degree-1 series of P^3 via logarithmic constraints, toric equivariance, and positivity. Additional results include that the PT series is a Laurent polynomial under sufficient positivity, a proof of the 2008 Oblomkov-Okounkov-Pandharipande conjecture on the capped vertex being a Laurent polynomial, and verification of the logarithmic DT/PT conjecture.

Significance. If the central reduction holds, this constitutes a substantial advance: the first complete verification of the GW/PT conjecture in the fully logarithmic setting with singular divisors, together with stronger structural results such as the Laurent polynomial property of the PT series. The reduction to a single explicit calculation (the degree-1 P^3 series) is a notable strength that makes the result falsifiable and computationally useful.

major comments (2)

[the reduction argument following the statement of the main theorem (as summarized in the abstract)] The load-bearing reduction asserting that logarithmic GW/PT constraints plus toric equivariance and positivity determine every primary invariant for arbitrary (including singular) ∂Y from the single degree-1 series of P^3 requires explicit verification. For singular ∂Y the logarithmic moduli spaces acquire additional strata and obstruction theories; the manuscript must show that no new independent classes appear and that the relations inherited from the smooth/P^3 case continue to hold without modification.
[the section establishing the Laurent polynomial property] The proof that the PT series is a Laurent polynomial under sufficient positivity (and the related 2008 capped-vertex conjecture) is stated as a stronger conclusion, but the precise positivity hypotheses and their compatibility with the singular fully-logarithmic obstruction theory need to be spelled out to confirm they do not introduce new denominators or poles.

minor comments (1)

[Introduction and notation] Clarify the precise definition of 'primary insertions' in the logarithmic setting when ∂Y is singular, to avoid ambiguity in the statement of the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the detailed comments. We address each major point below and indicate the revisions we plan to incorporate.

read point-by-point responses

Referee: The load-bearing reduction asserting that logarithmic GW/PT constraints plus toric equivariance and positivity determine every primary invariant for arbitrary (including singular) ∂Y from the single degree-1 series of P^3 requires explicit verification. For singular ∂Y the logarithmic moduli spaces acquire additional strata and obstruction theories; the manuscript must show that no new independent classes appear and that the relations inherited from the smooth/P^3 case continue to hold without modification.

Authors: The reduction proceeds from the uniform construction of the logarithmic moduli spaces and their obstruction theories via the logarithmic structure on the pair (Y|∂Y). This construction applies identically whether ∂Y is smooth or singular, with the additional strata in the singular case already incorporated into the logarithmic virtual class. The toric equivariance and positivity constraints then generate the same system of relations in all cases, as these relations arise from the action on the ambient toric variety and the positivity of the curve class with respect to the divisor components; they do not depend on smoothness of ∂Y. Consequently, no new independent classes appear. To address the request for explicit verification, we will insert a dedicated subsection that compares the obstruction theories in the smooth and singular settings and confirms that the inherited relations remain unmodified. revision: yes
Referee: The proof that the PT series is a Laurent polynomial under sufficient positivity (and the related 2008 capped-vertex conjecture) is stated as a stronger conclusion, but the precise positivity hypotheses and their compatibility with the singular fully-logarithmic obstruction theory need to be spelled out to confirm they do not introduce new denominators or poles.

Authors: The positivity hypotheses are the requirement that the curve class intersects every component of ∂Y positively; under this condition the equivariant localization formula for the PT series produces only non-negative powers in the relevant variables, yielding a Laurent polynomial. The same formula applies verbatim in the fully logarithmic setting because the virtual dimension and the weights in the obstruction theory are unchanged by the singularities of ∂Y. We will expand the relevant section to state the hypotheses explicitly, add a short compatibility remark with the singular obstruction theory, and confirm that no new denominators arise. revision: yes

Circularity Check

0 steps flagged

No circularity: toric pair invariants reduced to independent P^3 series via logarithmic constraints

full rationale

The paper's central claim is that logarithmic GW/PT constraints together with toric equivariance and positivity determine all primary invariants for any toric pair (Y|∂Y) from the single degree-1 series of P^3. This reduction is derived from the structure of the logarithmic theory rather than by fitting parameters to the target data or by self-definition. The degree-1 P^3 series is treated as an external, independently computable input, and the proof for singular ∂Y extends the same constraint framework without introducing new fitted elements or renaming known results. Self-citations, such as to the 2008 OOPP conjecture proved as a byproduct, are not load-bearing for the main correspondence. The derivation chain therefore remains self-contained against external benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard background results in logarithmic Gromov-Witten and DT/PT theory for toric varieties; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of toric varieties, logarithmic structures, and equivariant cohomology
Invoked to reduce computations to the degree-1 series of P^3.

pith-pipeline@v0.9.0 · 5506 in / 1160 out tokens · 37453 ms · 2026-05-15T14:34:06.977529+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Using the constraints of the logarithmic theory, the complete evaluation of toric pairs is determined by a single calculation — the degree 1 series of P^3.

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

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