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arxiv: 2605.30008 · v1 · pith:QOAMBXIOnew · submitted 2026-05-28 · 🧮 math.AG

The multiple cover formula for K3 and abelian surfaces

Georg Oberdieck , Rahul Pandharipande This is my paper

Pith reviewed 2026-06-29 00:30 UTC · model grok-4.3

classification 🧮 math.AG
keywords K3 surfacesabelian surfacesGromov-Witten invariantsstable pairsmultiple cover formularelative threefoldsGW/PT correspondence
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The pith

The multiple cover formulas for reduced descendent Gromov-Witten invariants of K3 and abelian surfaces in imprimitive classes follow from the conjectural families GW/PT correspondence for semipositive relative 3-folds with primary insertion

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

The paper shows that the multiple cover formulas conjectured earlier for K3 and abelian surfaces can be derived from a single assumption on the families GW/PT correspondence. This assumption concerns the equality of certain generating functions on the Gromov-Witten and Pandharipande-Thomas sides for the relative threefold obtained by taking the surface times a line with two marked sections. A reader cares because the formulas reduce all calculations in imprimitive classes to the already-solved primitive case, completing the determination of these invariants. The argument proceeds by rewriting the multiple cover relation as a statement about a localization vertex on the Gromov-Witten side, then moving the statement across the conjectural correspondence to the stable pairs side where it is verified directly.

Core claim

The multiple cover formula for a surface S is equivalent to a property of an appropriate localization vertex in the relative Gromov-Witten theory of the threefold S times the projective line with the two sections removed. The families GW/PT correspondence transfers this vertex property from the Gromov-Witten side to the stable pairs side. On the stable pairs side the formula is established geometrically by cosection localization and universality. As an intermediate step a DT/PT correspondence is proved for the reduced theories of the same relative threefolds by wall-crossing.

What carries the argument

the localization vertex in the relative Gromov-Witten theory of (S × ℙ¹ / S₀ ∪ S∞)

If this is right

  • The multiple cover formulas hold on the stable pairs side once the correspondence is granted.
  • All reduced descendent invariants of K3 and abelian surfaces in any curve class become computable from the primitive-class data.
  • A DT/PT correspondence holds for the reduced theories of the relative threefolds (S × ℙ¹ / S₀ ∪ S∞).

Where Pith is reading between the lines

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

  • The same vertex-recasting step could be attempted for other surfaces once analogous multiple cover conjectures are formulated.
  • A direct verification of the families GW/PT correspondence on even one such relative threefold would immediately yield the multiple cover formulas for that surface.

Load-bearing premise

The conjectural families GW/PT correspondence holds for the semipositive relative 3-folds (S × ℙ¹ / S₀ ∪ S∞) with primary insertions.

What would settle it

An explicit computation, for a fixed imprimitive class on a K3 surface, showing that the stable pairs multiple cover formula fails while the families GW/PT correspondence holds for the corresponding relative threefold, or vice versa.

read the original abstract

All reduced descendent Gromov-Witten invariants of $K3$ and abelian surfaces in primitive curve classes can be calculated by the methods of \cite{BOPY,MPT}. To handle the imprimitive curve classes, a multiple cover formula was conjectured in \cite{ObPand} for $K3$ surfaces and in \cite{O_NLGW} for abelian surfaces. We prove here that both descendent multiple cover formulas are implied by the conjectural families GW/PT correspondence for semipositive relative 3-folds with primary insertions. The implication is proven by showing that the multiple cover formula for $S$ can be recast as a property of an appropriate localization vertex for the relative 3-fold Gromov-Witten theory of $(S\times \mathbb{P}^1/S_0 \cup S_\infty)$. The families GW/PT correspondence then transfers the multiple cover formula from the Gromov-Witten side to the stable pairs side where the formula is proven geometrically by studying cosections and applying universality properties. Along the way, we prove a DT/PT correspondence for the reduced theories of $(S\times \mathbb{P}^1/S_0 \cup S_\infty)$ using the wallcrossing techniques of Kuhn-Liu-Thimm \cite{KLT2,KLT}.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that the conjectured descendent multiple cover formulas for reduced Gromov-Witten invariants of K3 and abelian surfaces in imprimitive classes follow from the conjectural families GW/PT correspondence for semipositive relative 3-folds with primary insertions. The argument recasts the multiple cover formula as a localization vertex property on the GW side of the relative 3-fold (S × ℙ¹ / S₀ ∪ S∞), transfers the property to the PT side via the correspondence, and verifies the resulting PT statement geometrically via cosections and universality. A reduced DT/PT correspondence for the same relative 3-fold is established en route using the wall-crossing techniques of Kuhn-Liu-Thimm.

Significance. If the families GW/PT correspondence holds, the result reduces the multiple cover formulas (previously conjectural in ObPand and O_NLGW) to a statement provable on the PT side, extending the primitive-class calculations of BOPY and MPT to imprimitive classes. The auxiliary reduced DT/PT correspondence via KLT wall-crossing is itself a concrete contribution to the relative theory of these surfaces. The conditional nature of the claim is stated explicitly and the logic is internally consistent.

minor comments (3)
  1. The abstract and introduction should explicitly flag that the main theorems are conditional on the families GW/PT conjecture (currently stated only in the body); this would clarify the logical status for readers.
  2. Notation for the relative 3-fold (S × ℙ¹ / S₀ ∪ S∞) and the precise meaning of 'primary insertions' should be recalled in §2 or §3 when the vertex property is defined, to avoid forward references.
  3. A short table or diagram summarizing the logical flow (GW vertex property → transfer → PT verification) would improve readability of the high-level argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and detailed summary of the manuscript, as well as for the recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proves that both descendent multiple cover formulas are implied by the conjectural families GW/PT correspondence for semipositive relative 3-folds with primary insertions. The argument recasts the formula as a localization vertex property on the GW side of (S × ℙ¹ / S₀ ∪ S∞), transfers the property via the external conjecture to the PT side, and establishes the resulting statement geometrically via cosections and universality; a reduced DT/PT correspondence is additionally shown by wall-crossing. No equation or claim reduces by construction to its own inputs, no fitted parameter is renamed as a prediction, and the central result is an explicit conditional implication rather than a self-referential loop. The cited conjecture is left open and does not serve as a load-bearing proven fact within the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unproven families GW/PT correspondence as the key external input; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The families GW/PT correspondence holds for semipositive relative 3-folds with primary insertions.
    Invoked to transfer the multiple cover formula from GW to PT side.

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