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Grothendieck topologies with logarithmic modifications

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Pith's one-line read This paper argues that the standard log étale topology is subtly broken and that m-type topologies, built by treating log blow-ups as covers, are the correct fix.

desk verdict Bold, mostly careful paper introducing m-topologies and claiming a fix to the full log étale site; Theorem C needs a specialist check, but the stress-test about log blow-ups not being monomorphisms does not survive contact with the log structures. read the letter →

arxiv 2510.23959 v2 pith:PLJ2NR2C submitted 2025-10-28 math.AG

Xianyu Hu , Maximilian Schimpf This is my paper · ORCID
classification math.AG MSC 14A21
keywords logarithmicgeometryGrothendiecktopologylogmodificationsm-étalemorphismsétalesitevaluativespaceuniversalsurjectivityblow-up
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Logarithmic invariants in geometry often survive log blow-ups, so a good topology for log schemes should treat log modifications as covers. This paper introduces m-type topologies—m-open, m-étale, m-smooth, m-fppf, and m-fpqc—in which log modifications, defined as universally surjective log étale monomorphisms, become covering maps. Along the way it identifies an error in the standard full log étale topology: a quasi-compact object in that site must have rank-≤1 log-characteristic stalks, which is far too restrictive and contradicts a published lemma; the paper repairs this by strengthening the definition of universal surjectivity so that it is tested against all integral log schemes. With that repair, the m-open topos of an fs log scheme is equivalent to the topos of its valuative log space, and sheaves on the m-type sites admit a simple characterization: descend along strict covers and along log blow-ups.

What carries the argument

The load-bearing objects are m-type morphisms and the strengthened notion of universal surjectivity. A log modification is declared to be a universally surjective log étale monomorphism, which automatically becomes proper under the new definition; m-étale morphisms are then defined by a chart criterion (or equivalently by a formal lifting criterion against arbitrary exact first-order thickenings). The technical core is Definition 5.6: a morphism of fs log schemes is universally surjective only if its base change along every integral log scheme is surjective, a strictly stronger requirement than the usual one. This redefinition makes the m-type sites functorial and makes log blow-ups into cov

What would settle it

Take an fs log scheme X with a point whose stalk M^gp_{X,x} has rank ≥ 2 and exhibit a single quasi-compact log étale cover (finite union) covering X; Theorem C says this is impossible, so a concrete such cover would disprove it. Alternatively, a log blow-up whose base change along some integral (non-fs) log scheme is not surjective would falsify Proposition 5.7(3) and the new universal surjectivity definition.

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Extended reading notes

Core claim

The central claim is that the standard full log étale topology is subtly broken and that a family of m-type topologies, with a corrected notion of universal surjectivity, is the right replacement. Concretely: in the small or big log étale site of an fs log scheme, an object is quasi-compact exactly when its underlying scheme is quasi-compact and every stalk of M^gp has rank ≤ 1; this contradicts a lemma in the published literature and implies the log étale site has too few quasi-compact objects, so results built on that lemma need revisiting. The paper's fix is to define universal surjectivity as surjectivity after all base changes along integral log schemes; under this definition log modifi

Load-bearing premise

The entire correction rests on the strengthened definition of universal surjectivity (Definition 5.6); if a log blow-up fails to be surjective after base change to some integral log scheme, or if this definition is too restrictive for the claimed proofs (the paper itself notes Proposition 2.6 depends on it), the repair of the log étale topology collapses, as does any reliance on the resolution theorem that reduces to Zariski log structures.

Editorial extensions

If this is right

  • The standard full log étale site has far fewer quasi-compact objects than previously believed, so the many papers using the classical log étale topology will need to check their quasi-compactness and descent arguments.
  • Any invariant that is invariant under log blow-ups automatically becomes a sheaf on the m-type sites; in particular log Gromov–Witten, log Chow, and log Hilbert theories all descend.
  • A presheaf is a sheaf on an m-type site if and only if it satisfies descent for strict type-covers and maps log blow-ups to isomorphisms—a drastically simplified sheaf condition.
  • Cohomology of any abelian sheaf on the m-open site vanishes above the logarithmic dimension, which is computed by an explicit formula involving point dimensions and log stalk ranks.
  • The m-open topos equivalence with the valuative log space topos gives a concrete spectral space model for the m-open site, enabling topological dimension arguments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the strengthened universal surjectivity is accepted, the m-type topologies likely agree with the 'dividing' topologies introduced in the motivic literature, giving a direct categorical foundation for logarithmic motives.
  • The redefinition effectively says that fs log schemes are not the correct test category for surjectivity; the broader category of integral log schemes is. This suggests that other log-geometric notions (e.g., properness, flatness) may also need to be reformulated with integral test objects.
  • The valuative-space equivalence suggests a new computational route for log étale cohomology: compute on the smaller, spectral valuative space rather than on the log scheme itself.
  • The global integralization theorem—every fs log scheme becomes Zariski-log after a log blow-up—could simplify many reduction arguments if it holds for the strengthened notion, since Zariski log structures are far easier to chart.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper introduces new Grothendieck topologies (m-open, m-étale, m-smooth, m-fppf, m-fpqc) on the category of fs log schemes. The key move is to declare log modifications — defined in Definition 1.1 as universally surjective log étale monomorphisms — to be covers, alongside strict scheme-theoretic covers. The authors prove a chart criterion for m-open morphisms, an infinitesimal lifting criterion for m-étale and m-smooth morphisms (Theorem A), characterize sheaves on all m-type sites (Theorem B), and establish an equivalence between the m-open topos and Kato's valuative log space (Theorem D), with a resulting cohomological vanishing criterion. The central new result is Theorem C: in the standard full log étale site, a quasi-compact object must have all stalks of M^gp of rank at most 1, contradicting a lemma in [28]. The paper then proposes a strengthened notion of universal surjectivity (Definition 5.6) to repair the log étale topology and asserts that this fixes [28, Lemma 3.14].

Significance. If Theorem C is correct, this is a substantial correction to the literature on the full log étale topology, with consequences for several papers built on Nakayama's work. The proposed m-type topologies are a genuinely new framework that may be useful for log invariance phenomena. The paper is generally carefully written: proofs are detailed, dependencies are often explicitly flagged (e.g., Remark 2.7, reliance on Nizioł's resolution theorem), and the main theorems are stated in a falsifiable form. The global integralization theorem and the valuative-space description are valuable independent contributions. The main risk is the strong contradiction with [28], which is asserted with a fairly terse proof, and the foundational monomorphism property of log blow-ups, which is cited rather than proved in the text.

major comments (2)
  1. [§1.1, Definition 1.1; §2.1, Proposition 2.6] The definition of logarithmic modification and the proof of Proposition 2.6 hinge on the assertion that every log blow-up is a monomorphism in fsLogSch, citing [33, Example 2.7] and [31, Theorem III.2.6.3(3)]. This is non-obvious: scheme-theoretically, Bl_0(A²)→A² is not a monomorphism. A naive counterexample using two points in the exceptional divisor does not automatically lift to fsLogSch because log morphisms carry additional unit data; in an exact chart the local coordinate e1/e2 is a unit, so distinct points can induce different log maps to the base. Nevertheless, since Definition 1.1 and the m-open topology rest entirely on this point, please include a proof, or at least state precisely which result in the cited references asserts this and explain why the scheme-theoretic intuition is misleading. This is load-bearing for Proposition 2.6 and for the claim that log blow-ups are m-op
  2. [§5.1, Proof of Theorem C] The proof of Theorem C is quite terse at the point where a finite subcover of the valuative cover of X is used to conclude that each étale chart U_i admits a quasi-compact log étale cover by valuative fs log schemes. The intended argument is presumably that pulling back the finite subcover along U_i→X gives a finite cover of U_i, but this should be written out explicitly. More importantly, the proof relies on [28, Lemma 3.11] and [28, Proposition 3.7] even though the paper claims that [28] is largely broken by Theorem C. Please verify that these cited results are independent of the allegedly erroneous Lemma 3.14, or supply alternative proofs. This matters because Theorem C is the paper's main correction.
minor comments (6)
  1. [§1.2, Definition 1.4] The phrase 'strict étale locally in T exactly one(at least one) dotted arrow' is missing a space/comma; also 'locally in T' is ambiguous — it means strict étale local on the thickening T.
  2. [§3.1, Lemma 3.2] In the statement, 'formally étale (resp. m-smooth)' should presumably read 'formally étale (resp. formally smooth)' for consistency with the proof.
  3. [§4, Proposition 4.3] The proof refers to Theorem 6.3 and Lemma 6.5 before they are stated. Add forward references or move the needed preliminaries earlier.
  4. [§6.1, Theorem D proof] 'form a topology base' should be 'form a base for the topology'.
  5. [§1.4, Theorem C discussion] The statements that Theorem C 'seems to break most of [28]' and 'impacts virtually every paper' are stronger than what is proved. I suggest tempering these claims or spelling out which specific results in [28] fail.
  6. [§5.2, Lemma 5.4] The equivalence of the three conditions in Lemma 5.4 is stated for 'an integral log scheme T' but the fiber products are taken in different categories (integral vs saturated). The notation in condition (3) is also a bit compressed; a short clarification would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found; the paper's claims are derived from explicit definitions and independent external facts, not from self-referential reductions.

full rationale

No circular reduction is exhibited in the paper. The central results (Theorems A–E, Proposition 4.3, Theorem C) are argued from explicit chart criteria, lifting properties, and structural lemmas in external references ([31] Ogus, [28] Nakayama, [29] Nizioł, [5] Achinger et al.), not from the paper's own conclusions. Theorem C is a statement about the standard log étale site; its proof uses Proposition 5.1 and lemmas from [28], and does not presuppose the paper's strengthened Definition 5.6. Definition 5.6 is an openly stipulated redefinition of universal surjectivity, and its consequences are then proved (Proposition 5.7, Proposition 4.3, Lemma 6.5), so it is not a fitted input masquerading as a prediction. Theorem B is a nontrivial refinement statement rather than a tautology. The only serious mathematical concern visible in the text is the assertion in Definition 1.1 that log blow-ups are log étale monomorphisms, supported by citations to [33] and [31]; if that assertion is false, the theory would be incorrect, but that is a correctness or support defect, not a self-referential reduction. The paper also flags its dependence on the strengthened universal surjectivity in Remark 2.7. No self-citation is load-bearing, and no equation or definition reduces to its own output by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No entities are invented and no parameters are fitted to data. The paper's central claims rest on two ad hoc definitions (log modification as universally surjective log étale monomorphism, and the strengthened universal surjectivity), one cited resolution theorem, and standard log-geometry background. The ad hoc definitions are the actual subject of the paper and are argued for, but they are the most delicate inputs.

assumptions (4)
  • ad hoc to paper A morphism of fs log schemes is a log modification iff it is a universally surjective log étale monomorphism (Definition 1.1).
    This definition is new to this paper (generalizing earlier notions from [15, 33]) and is the foundation for all m-type topologies; its correctness is what the paper argues for.
  • ad hoc to paper Universal surjectivity of morphisms of fs log schemes is tested on base changes along arbitrary integral log schemes (Definition 5.6).
    The paper introduces this stronger notion to fix the issues in the log étale site; it is used in Proposition 2.6 and the global integralization theorem. This is a structural departure from the standard definition and is load-bearing.
  • domain assumption Every fs log scheme with étale log structure admits a log blow-up whose log structure is Zariski ([29, Theorem 5.4]).
    Used to reduce to Zariski log structures in proofs of Theorem B and D; if this resolution theorem failed, the topos equivalence and sheaf characterization would not follow in the stated generality.
  • standard math Standard fine-saturated log geometry facts from Ogus [31] (chart lemmas, integrality, saturation, Zariski log structure identification) are used freely.
    The paper relies on the standard references for log geometry; these are unproved background results.

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Cite this review

Pith. "Pith review of Grothendieck topologies with logarithmic modifications." pith.science (2026). https://pith.science/paper/PLJ2NR2C

@misc{pith2026251023959,
  author       = {Pith},
  title        = {Pith review of: Grothendieck topologies with logarithmic modifications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PLJ2NR2C}},
  note         = {Machine review of arXiv:2510.23959}
}
read the original abstract

Many concepts in log geometry are invariant under log blow-ups. To formalize this invariance, we introduce the m-open, m-\'etale, m-smooth, m-fppf, and m-fpqc topologies for fs log schemes. These refine the standard topologies from scheme theory by treating abstract log modifications as covers. For example, the m-\'etale topology is a subtopology of full log \'etale topology, characterized by a stronger lifting property than for log \'etale maps. Along the way, we identify and correct an error in the definition of the full log \'etale topology. We also prove a global integralization theorem by logarithmic blow-ups and use it to describe the m-open topos as a limit over log blow-ups. Finally, we characterize the sheaves on the m-type sites and connect the m-open site to Kato's valuative space.

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Forward citations

Cited by 1 Pith paper

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