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arxiv: 2605.01730 · v1 · submitted 2026-05-03 · 🧮 math.AG · math-ph· math.MP

Fixed point locus of Moduli spaces of Sheaves on Toric DM stacks

Promit Kundu This is my paper

Pith reviewed 2026-05-09 17:16 UTC · model grok-4.3

classification 🧮 math.AG math-phmath.MP
keywords torsion-free sheavestoric DM stacksmoduli spacesfixed point locuscharacteristic functionsGieseker stabilitycombinatorial descriptiontoric varieties
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The pith

The torus fixed point locus in the moduli space of stable sheaves on a smooth toric DM stack is described explicitly by characteristic functions.

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

This paper extends combinatorial descriptions of toric sheaves to smooth toric Deligne-Mumford stacks in any dimension. It shows that the natural action of the torus on the stack lifts to the moduli space of torsion-free sheaves that are stable in a modified Gieseker sense. The fixed points under this lifted action are then classified by characteristic functions, which are combinatorial invariants associated to the sheaves. This setup recovers previous results for ordinary toric varieties and opens the way to calculating topological invariants of these moduli spaces.

Core claim

Extending work of Klyachko, Perling and Kool we develop a combinatorial description of torsion free toric sheaves in any dimension on smooth toric DM stacks. We investigate their basic properties and under certain conditions recover some known results on smooth toric varieties. The action of the torus lifts to the moduli of torsion free modified Gieseker stable sheaves on the smooth DM stack, and we express its fixed point locus explicitly in terms of certain finer invariants called characteristic functions. These techniques will be exploited to compute topological invariants of the moduli of modified stable torsion free sheaves on smooth toric DM stacks.

What carries the argument

Characteristic functions, the combinatorial invariants attached to torsion-free toric sheaves that parametrize the torus-fixed points inside the moduli space.

If this is right

  • The fixed point locus can be used to compute topological invariants such as Euler characteristics of the moduli spaces via localization.
  • Known results on smooth toric varieties are recovered as special cases of the stacky description.
  • The combinatorial data of characteristic functions applies uniformly in any dimension for smooth toric DM stacks.
  • Basic properties of torsion-free toric sheaves on these stacks are established through their combinatorial presentation.

Where Pith is reading between the lines

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

  • The explicit fixed-locus description may make it feasible to calculate previously inaccessible invariants for specific stacky examples.
  • The lifting of the torus action to the moduli space suggests analogous fixed-point techniques could apply to other moduli problems involving sheaves or complexes on toric stacks.
  • Connections between combinatorial sheaf data and enumerative invariants on orbifolds become more direct when the fixed loci are parametrized this way.

Load-bearing premise

A combinatorial description of torsion-free toric sheaves exists in any dimension on smooth toric DM stacks and the torus action lifts to the moduli space under the stated stability conditions.

What would settle it

An explicit enumeration of torus-fixed stable sheaves on a concrete low-dimensional smooth toric DM stack that fails to match the set of characteristic functions predicted by the combinatorial description.

read the original abstract

Extending work of Klyachko, Perling and Kool we develop a combinatorial description of torsion free toric sheaves in any dimension on smooth toric DM stacks. We investigate their basic properties and under certain conditions recover some known results on smooth toric varieties. The action of the torus lifts to the moduli of torsion free modified Gieseker stable sheaves on the smooth DM stack, and we express its fixed point locus explicitly in terms of certain finer invariants called characteristic functions. These techniques will be exploited to compute topological invariants of the moduli of modified stable torsion free sheaves on smooth toric DM stacks.

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 manuscript extends the combinatorial description of torsion-free toric sheaves, previously developed by Klyachko, Perling, and Kool for smooth toric varieties, to smooth toric Deligne-Mumford stacks in arbitrary dimension. It studies basic properties of these sheaves, recovers known results for varieties under suitable conditions, asserts that the torus action lifts to the moduli space of torsion-free modified Gieseker stable sheaves, and expresses the fixed-point locus explicitly via characteristic functions. The techniques are positioned to compute topological invariants of these moduli spaces.

Significance. If the combinatorial description is complete and the lifting of the torus action is rigorously justified, the work would provide a concrete combinatorial handle on fixed loci and invariants for moduli spaces on toric DM stacks, extending classical results to a setting that includes orbifold singularities. The introduction of characteristic functions as finer invariants could facilitate explicit calculations that are otherwise inaccessible.

major comments (1)
  1. [section on lifting of the torus action and fixed-locus description] The central claim that the torus action lifts to the moduli space of modified Gieseker stable sheaves (abstract and the section introducing the fixed-locus description) requires that modified Gieseker stability be invariant under pullback by torus elements. The manuscript does not supply an explicit verification that the Hilbert polynomial comparisons and subsheaf stability conditions remain unchanged when the multi-filtration data of the characteristic functions are transformed by the torus action on a higher-dimensional DM stack; this invariance is load-bearing for the subsequent explicit description of the fixed locus.
minor comments (2)
  1. [abstract and introduction] The abstract states that known results on smooth toric varieties are recovered 'under certain conditions' but does not list the precise conditions or recovered statements; this should be made explicit in the introduction or a dedicated subsection.
  2. Notation for the characteristic functions and the modified Gieseker stability condition should be introduced with a clear comparison table to the classical Klyachko/Perling data to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of rigorously justifying the lifting of the torus action. We address the major comment below and will incorporate the suggested clarification in the revised version.

read point-by-point responses

  1. Referee: [section on lifting of the torus action and fixed-locus description] The central claim that the torus action lifts to the moduli space of modified Gieseker stable sheaves (abstract and the section introducing the fixed-locus description) requires that modified Gieseker stability be invariant under pullback by torus elements. The manuscript does not supply an explicit verification that the Hilbert polynomial comparisons and subsheaf stability conditions remain unchanged when the multi-filtration data of the characteristic functions are transformed by the torus action on a higher-dimensional DM stack; this invariance is load-bearing for the subsequent explicit description of the fixed locus.

    Authors: We agree that an explicit verification of the invariance of modified Gieseker stability under torus pullbacks is necessary to support the lifting claim and the subsequent fixed-locus description, especially in the higher-dimensional DM stack setting where the multi-filtration data of the characteristic functions becomes more intricate. While the combinatorial description in the manuscript (extending Klyachko-Perling-Kool) provides the framework for how torus elements act on the filtrations, we acknowledge that a direct check of the stability conditions was not spelled out. In the revised manuscript we will add a dedicated paragraph or short subsection immediately preceding the fixed-locus theorem. This addition will: (i) recall the precise definition of modified Gieseker stability via Hilbert polynomials and subsheaf inequalities; (ii) describe the explicit transformation rule for the multi-filtration data (characteristic functions) induced by a torus element; and (iii) verify that the transformed data preserve both the Hilbert polynomial comparisons and the stability inequalities for subsheaves. The argument will rely only on the already-established properties of the combinatorial description and will not require new results. We believe this clarification will make the lifting statement fully rigorous while leaving the main theorems unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends external literature independently

full rationale

The paper extends combinatorial descriptions from Klyachko, Perling, and Kool to smooth toric DM stacks, introducing characteristic functions as new invariants to describe the fixed locus of the torus action on moduli of modified Gieseker stable sheaves. No steps reduce by construction to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations; the lifting of the action and explicit expression via invariants follow from the developed combinatorial setup under the stated stability conditions, remaining self-contained against external benchmarks without tautological reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review prevents exhaustive ledger; the work relies on standard background from toric geometry and sheaf theory rather than introducing new free parameters or invented entities.

axioms (2)
  • domain assumption Smooth toric DM stacks admit a combinatorial description of their torsion-free toric sheaves
    Invoked to extend Klyachko-Perling-Kool results to any dimension.
  • domain assumption The torus action lifts to the moduli space of modified Gieseker stable sheaves
    Central to expressing the fixed locus.

pith-pipeline@v0.9.0 · 5395 in / 1394 out tokens · 39887 ms · 2026-05-09T17:16:21.578033+00:00 · methodology

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Reference graph

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