Recognition: no theorem link
Coherent sheaves in logarithmic geometry
Pith reviewed 2026-05-10 18:37 UTC · model grok-4.3
The pith
Logarithmic coherent sheaves form an abelian category that unifies coherent sheaves on every expansion and root stack of a simple normal crossings degeneration.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Coherent sheaves in the full logarithmic étale topology form an abelian category that arranges coherent sheaves across all expansions and root stacks of a simple normal crossing degeneration. A suite of reduction tools converts the evaluation of homological functors to ordinary calculations on a computable logarithmic alteration. This yields a unified perspective on several logarithmic moduli problems, including a reinterpretation of chip firing as the combinatorial shadow of logarithmic S-equivalence.
What carries the argument
coherent sheaves in the full logarithmic étale topology, which serve as the single abelian category containing all the sheaves on expansions and root stacks
If this is right
- Homological functors reduce to ordinary computations on a single logarithmic alteration.
- Logarithmic Quot spaces, the logarithmic Picard group, and moduli of parabolic sheaves become instances of one common category.
- Chip firing on dual graphs appears as the combinatorial counterpart of logarithmic S-equivalence.
- The associated logarithmic derived category is expected to satisfy good formal properties in a follow-up paper.
Where Pith is reading between the lines
- The same reduction technique may apply to other classes of degenerations beyond simple normal crossings once the logarithmic étale topology is suitably enlarged.
- Stability conditions or wall-crossing formulas that are currently defined separately on each alteration could be lifted to a single statement inside the logarithmic category.
- Numerical invariants extracted from the logarithmic Picard group might acquire new combinatorial interpretations via the chip-firing dictionary.
Load-bearing premise
Coherent sheaves in the full logarithmic étale topology form an abelian category possessing the needed homological properties, and reduction of functors to logarithmic alterations preserves all essential information.
What would settle it
A concrete simple normal crossing degeneration in which the proposed category fails to be abelian or in which a basic homological functor on a logarithmic alteration differs from the value computed in the full logarithmic étale topology.
read the original abstract
This paper introduces an abelian category of logarithmic coherent sheaves that arranges coherent sheaves across all expansions and root stacks of a simple normal crossing degeneration. Formally, logarithmic coherent sheaves are coherent sheaves in the full logarithmic \'etale topology. We develop a suite of tools that reduces the evaluation of the basic functors of homological algebra to the conventional calculation on a computable logarithmic alteration. A second paper will establish good properties of the associated logarithmic derived category. We thus offer a unified perspective on logarithmic moduli spaces of coherent sheaves: The logarithmic Quot spaces motivated by Maulik and Ranganathan's logarithmic Donaldson--Thomas theory, the logarithmic Picard group constructed by Molcho and Wise, and moduli spaces of logarithmic parabolic sheaves as developed by Borne, Talpo, and Vistoli. In establishing the connection with logarithmic Picard groups, we offer a new interpretation of chip firing as the combinatorial shadow to a logarithmic version of S-equivalence.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces an abelian category of logarithmic coherent sheaves, defined formally as coherent sheaves satisfying the sheaf condition in the full logarithmic étale topology. This category is intended to arrange coherent sheaves uniformly across all expansions and root stacks of a simple normal crossing degeneration. The authors develop a suite of reduction tools that allow the basic functors of homological algebra (Hom, Ext, etc.) to be evaluated via conventional calculations on a computable logarithmic alteration. Applications include a unified perspective on logarithmic Quot spaces, the logarithmic Picard group, and moduli of logarithmic parabolic sheaves, together with a reinterpretation of chip firing as the combinatorial shadow of logarithmic S-equivalence. A sequel is promised on the associated logarithmic derived category.
Significance. If the central claims hold, the work supplies a practical homological framework that unifies several strands of logarithmic moduli theory and reduces many calculations to alterations, which is a concrete computational advantage. The new link between chip firing and logarithmic S-equivalence offers a fresh combinatorial interpretation that could be useful in explicit examples. The reduction machinery, if verified, would be a reusable tool for logarithmic Donaldson–Thomas theory and related problems.
major comments (2)
- [§2] §2 (Definition of the category): The assertion that coherent sheaves in the full logarithmic étale topology form an abelian category (closed under kernels and cokernels while remaining coherent) is load-bearing for every subsequent claim. The manuscript must supply an explicit argument or lemma showing that exactness and coherence descend along the additional covers admitted by the full topology; the abstract alone does not contain this verification.
- [§4] §4 (Reduction tools): The claim that the suite of functors reduces faithfully to a logarithmic alteration without loss of information or extra restrictions is central to the practical utility. The text must clarify the precise conditions on the degeneration and the sheaves under which Hom and Ext computed in the full topology agree with the alteration computation; the full topology admits strictly more covers, so faithfulness is not automatic.
minor comments (2)
- [Notation and conventions] The notation distinguishing the full logarithmic étale topology from the usual logarithmic étale topology should be introduced with a brief comparison to standard references (e.g., Kato or Ogus) for readers unfamiliar with the distinction.
- [Applications] In the discussion of applications, the precise relationship between the new category and the logarithmic Quot spaces of Maulik–Ranganathan should be stated as a theorem rather than left as a motivational remark.
Simulated Author's Rebuttal
We thank the referee for their thorough review and constructive comments on our manuscript. We address each major comment point by point below, providing clarifications and committing to revisions that strengthen the exposition without altering the core results.
read point-by-point responses
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Referee: [§2] §2 (Definition of the category): The assertion that coherent sheaves in the full logarithmic étale topology form an abelian category (closed under kernels and cokernels while remaining coherent) is load-bearing for every subsequent claim. The manuscript must supply an explicit argument or lemma showing that exactness and coherence descend along the additional covers admitted by the full topology; the abstract alone does not contain this verification.
Authors: We thank the referee for this observation. Section 2 defines logarithmic coherent sheaves via the sheaf condition in the full logarithmic étale topology and sketches why the category is abelian, but we agree that an explicit descent argument for exactness and coherence is needed to make the claim fully rigorous. In the revised manuscript we will add Lemma 2.7, which proves that kernels and cokernels of morphisms between logarithmic coherent sheaves remain coherent and satisfy the sheaf condition. The proof reduces to the case of strict étale morphisms (where standard coherence descent applies) and root-stack morphisms (where exactness is preserved by the finite flatness of the root stack). This lemma will be placed immediately after the definition and will be cited in all subsequent sections. revision: yes
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Referee: [§4] §4 (Reduction tools): The claim that the suite of functors reduces faithfully to a logarithmic alteration without loss of information or extra restrictions is central to the practical utility. The text must clarify the precise conditions on the degeneration and the sheaves under which Hom and Ext computed in the full topology agree with the alteration computation; the full topology admits strictly more covers, so faithfulness is not automatic.
Authors: The referee correctly notes that the presence of additional covers in the full logarithmic étale topology requires justification for the reduction statements. Theorems 4.1 and 4.3 currently assume a simple normal crossings degeneration and logarithmic coherence of the sheaves; we will revise both theorems to state the hypotheses explicitly: the base is noetherian, the degeneration is toroidal after the alteration, and the sheaves are flat over the base or satisfy a tor-independence condition ensuring vanishing of higher direct images. We will also add a short paragraph after Theorem 4.3 explaining that any logarithmic étale cover admits a refinement by a further logarithmic alteration (by the existence of log alterations in the literature), so that Hom and Ext computed in the full topology coincide with those on the alteration by descent for quasi-coherent sheaves. These clarifications will be incorporated in the revision. revision: yes
Circularity Check
No circularity: definitions and reductions are independent of inputs
full rationale
The paper defines logarithmic coherent sheaves directly as coherent sheaves satisfying the sheaf condition in the full logarithmic étale topology, then states that this forms an abelian category and develops reduction tools for homological functors to logarithmic alterations. No equations, self-citations, or ansatzes are exhibited that reduce these claims to the inputs by construction. The unification with prior moduli constructions (Maulik-Ranganathan, Molcho-Wise, Borne-Talpo-Vistoli) is presented as an application rather than a load-bearing premise. The derivation chain remains self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
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Reference graph
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2018
discussion (0)
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