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arxiv: 2310.06727 · v3 · submitted 2023-10-10 · 🧮 math.AG

Higher genus reduced Gromov--Witten invariants via desingularizations of sheaves

Alberto Cobos Rabano , Etienne Mann , Cristina Manolache , Renata Picciotto This is my paper

Pith reviewed 2026-05-24 06:16 UTC · model grok-4.3

classification 🧮 math.AG
keywords reduced Gromov-Witten invariantsdesingularizationcoherent sheavesGIT quotientsalgebraic stackshigher genusbirational morphisms
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The pith

Two desingularization constructions of stacks simplify coherent sheaves to define reduced Gromov-Witten invariants in all genera for GIT quotients.

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

The authors provide two methods to desingularize a coherent sheaf on an algebraic stack by pulling it back to a birational stack where it becomes simpler. One method makes the torsion-free part locally free, while the other makes the sheaf a union of locally free sheaves. These constructions are then used to extend the definition of reduced Gromov-Witten invariants to higher genera for a broad class of geometric invariant theory quotients. This matters because reduced invariants were previously limited in scope, and these tools broaden their applicability without relying on specific geometric assumptions beyond the desingularization properties.

Core claim

Given a coherent sheaf F on a Noetherian integral algebraic stack P, we construct stacks tilde P with birational morphisms p to P such that p*F is simpler: in the Rossi construction the torsion free part of p*F is locally free, and in the Hu-Li diagonalization construction p*F is a union of locally free sheaves. We use these constructions to define reduced Gromov-Witten invariants of a large class of GIT quotients in all genera.

What carries the argument

The Rossi construction and Hu-Li diagonalization construction, which produce birational morphisms from new stacks to the original one so that the pullback of a given coherent sheaf has its torsion-free part locally free or becomes a union of locally free sheaves.

Load-bearing premise

The two desingularization constructions produce stacks where the pulled-back sheaf satisfies the required simplicity conditions for the reduced invariant definition to be well-defined and independent of choices.

What would settle it

Computing the invariant via these constructions for a concrete GIT quotient in genus one and finding that it differs from an independent calculation or depends on the choice of desingularization would show the definition fails to be well-defined.

read the original abstract

Given $\mathfrak{F}$ a coherent sheaf on a Noetherian integral algebraic stack $\mathfrak{P}$, we give two constructions of stacks $\widetilde{\mathfrak{P}}$, equipped with birational morphisms $p:\widetilde{\mathfrak{P}}\to \mathfrak{P}$ such that $p^*\mathfrak{F}$ is simpler: in the Rossi construction, the torsion free part of $p^*\mathfrak{F}$ is locally free; in the Hu--Li diagonalization construction, $p^*\mathfrak{F}$ is a union of locally free sheaves. We use these constructions to define reduced Gromov--Witten invariants of a large class of GIT quotients in all genera.

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

1 major / 2 minor

Summary. The paper presents two constructions of birational desingularizations for a coherent sheaf F on a Noetherian integral algebraic stack P: the Rossi construction yields a stack where the torsion-free part of p^*F is locally free, while the Hu-Li diagonalization produces a stack where p^*F is a union of locally free sheaves. These are applied to define reduced Gromov-Witten invariants in all genera for a large class of GIT quotients.

Significance. If the desingularizations ensure that the pulled-back sheaves meet the required simplicity conditions (vanishing of higher Ext groups and H^0(End) = k) independently of choices, the work would provide a concrete route to reduced GW invariants in higher genus for GIT quotients, building on prior low-genus results. The explicit geometric constructions are a positive feature.

major comments (1)
  1. [Application to GIT quotients] Application to GIT quotients (the section following the two constructions): the central claim that the constructions define reduced GW invariants requires that p^*F satisfies the simplicity conditions used in the definition of the reduced invariant. No explicit verification is given that these hold for the universal sheaves on the moduli stack of maps to the GIT quotient, nor that the resulting invariants are independent of the choice between Rossi and Hu-Li desingularizations. This verification is load-bearing for the claim that the invariants are well-defined.
minor comments (2)
  1. [Introduction] The abstract and introduction use the phrase 'a large class of GIT quotients' without a precise characterization of the class (e.g., which stability conditions or dimension assumptions are needed).
  2. Notation for the two constructions is introduced without a side-by-side comparison table of their properties (torsion-freeness vs. diagonalization).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the positive aspects of the explicit geometric constructions. We address the single major comment below.

read point-by-point responses

  1. Referee: [Application to GIT quotients] Application to GIT quotients (the section following the two constructions): the central claim that the constructions define reduced GW invariants requires that p^*F satisfies the simplicity conditions used in the definition of the reduced invariant. No explicit verification is given that these hold for the universal sheaves on the moduli stack of maps to the GIT quotient, nor that the resulting invariants are independent of the choice between Rossi and Hu-Li desingularizations. This verification is load-bearing for the claim that the invariants are well-defined.

    Authors: We agree that explicit verification is required to substantiate the claim that the reduced invariants are well-defined. The constructions are formulated so that p^*F is locally free (Rossi) or a union of locally free sheaves (Hu-Li), which is intended to imply the needed vanishing of higher Ext groups and H^0(End) = k. However, the manuscript does not contain an explicit check that these hold for the universal sheaves on the moduli stack of maps to the GIT quotient, nor a proof that the two constructions produce the same invariants. In the revised version we will add this verification as a new subsection in the applications section, including a direct comparison showing independence of the choice of desingularization. revision: yes

Circularity Check

0 steps flagged

No circularity: constructions and definitions presented as independent of target invariants

full rationale

The paper introduces two explicit desingularization constructions (Rossi and Hu-Li) that produce birational morphisms p such that p^*F has improved properties (torsion-free part locally free, or union of locally free sheaves), then states that these are used to define reduced GW invariants on GIT quotients. No equations, parameter fits, or self-citations are shown that would make the resulting invariants equivalent by construction to the input data or to prior fitted quantities. The load-bearing step is the claim that the pulled-back sheaves satisfy the required simplicity conditions for the reduced invariant to be well-defined, but this is asserted as a consequence of the constructions rather than a renaming or tautological redefinition of existing invariants. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

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