Removing the Torsion-free Hypothesis in a Positivity Theorem on Deligne-Mumford Stacks

Jihao Liu; Shengyu Hou

Every coherent quotient of a log cotangent power on a smooth proper DM stack has pseudo-effective first Chern class once a big-determinant subsheaf appears.

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2026-07-30 14:46 UTC pith:L7SEMNKF

load-bearing objection Short, correct note that closes the torsion gap left open in CMZ and cleans their positivity theorem; elementary but real.

arxiv 2607.26989 v1 pith:L7SEMNKF submitted 2026-07-29 math.AG

Removing the Torsion-free Hypothesis in a Positivity Theorem on Deligne-Mumford Stacks

Shengyu Hou , Jihao Liu This is my paper

classification math.AG MSC 14A2014C1714F1014J17

keywords Deligne–Mumford stacklogarithmic cotangent sheafpseudo-effective divisorfirst Chern classtorsion subsheafpositivitynormal crossings

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

A prior positivity theorem on smooth proper Deligne–Mumford stacks said that if some tensor power of the logarithmic cotangent sheaf contains a subsheaf with big determinant, then every torsion-free coherent quotient of every positive tensor power has pseudo-effective first Chern class. The original authors left open whether the torsion-free restriction was essential, because first Chern classes of torsion sheaves were only defined when those sheaves arise as quotients of torsion-free sheaves. This paper proves that the torsion subsheaf of any coherent quotient of a locally free sheaf is itself such a quotient, so its first Chern class is well-defined and effective. Adding the two pieces removes the hypothesis: every coherent quotient, torsion or not, has pseudo-effective first Chern class. The result answers the original authors’ question and makes the positivity statement apply to all coherent quotients that arise in practice.

Core claim

On an integral Deligne–Mumford stack, if E is locally free and π: E ↠ Q is any coherent quotient, then the preimage E′ = π⁻¹(Q_tor) is torsion-free and still surjects onto the torsion subsheaf Q_tor. Consequently c₁(Q_tor) is defined, equals an effective codimension-one cycle by the length formula, and adding it to the already pseudo-effective class of the torsion-free quotient yields that c₁(Q) itself is pseudo-effective. This removes the torsion-free hypothesis from the Casalaina-Martin–Zhjeqi positivity theorem.

What carries the argument

The preimage construction E′ := π⁻¹(Q_tor) together with the snake lemma: because E is locally free (hence torsion-free), E′ is torsion-free and the restricted map E′ ↠ Q_tor is surjective, so Q_tor becomes a quotient of a torsion-free sheaf and inherits a well-defined first Chern class.

Load-bearing premise

The whole argument rests on the paper’s chosen definition of first Chern class for coherent sheaves that are quotients of torsion-free sheaves, and on the length formula that makes the torsion contribution effective.

What would settle it

Exhibit a smooth proper integral DM stack with projective coarse space, a normal-crossings divisor Δ, and a coherent quotient Q of some tensor power of Ω¹(log Δ) whose first Chern class (computed in the paper’s sense) fails to be pseudo-effective, while some positive tensor power still contains a subsheaf with big determinant.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The positivity theorem now applies to arbitrary coherent quotients, not only torsion-free ones.
c₁ of any torsion subsheaf arising this way is an effective codimension-one cycle and can be added freely.
Numerical constraints coming from big-determinant subsheaves of log cotangent powers constrain every coherent quotient of those powers.
Later arguments that previously had to reduce to the torsion-free case can work directly with the original quotient.

Where Pith is reading between the lines

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

The same preimage argument should let other stacky positivity statements drop torsion-free hypotheses whenever the ambient sheaf is locally free.
Once c₁ is defined for all such quotients, one can ask whether higher Chern classes or discriminant inequalities also extend without torsion-freeness.
The length formula suggests that the torsion contribution is supported exactly on the codimension-one locus where the quotient fails to be locally free, giving a concrete geometric interpretation of the error term.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Short, correct note that closes the torsion gap left open in CMZ and cleans their positivity theorem; elementary but real.

read the letter

This is a three-page note that does exactly what the title says. Casalaina-Martin–Zhjeqi left open whether the torsion subsheaf of a coherent quotient of a locally free sheaf on an integral DM stack is itself a quotient of something torsion-free, so that c1 is defined in their conventions. Hou–Liu prove it is: the preimage under the surjection is torsion-free (local freeness + subsheaf of torsion-free) and still surjects (snake lemma in the abelian category of coherent sheaves). Once that is in hand, additivity plus their length formula for c1 of torsion makes the torsion contribution effective, the torsion-free quotient is already covered by the original theorem, and the hypothesis drops. That is the whole paper.

What is new is precisely that observation (Theorem A) and the cleaned statement (Theorem B). It is elementary sheaf theory, not a new method, but it answers an explicit remark in the source paper and yields the cleaner positivity theorem people will actually want to cite. The write-up is clear, the lemmas are standard, and the argument does not circle. The Chern-class conventions are taken from their companion note; if you already work in that setup there is no mismatch, and the local DVR/Smith computation is the usual one. AI disclosure is present and irrelevant to the math.

Soft spots are minor and proportional: the result is narrow, significance is local to positivity on DM stacks, and there is no formal verification. None of that undermines the claim. The central argument holds.

This is for people already reading the CMZ series or working with logarithmic cotangent sheaves on stacks. It deserves a serious referee rather than a desk reject; the note is short, correct, and useful inside its lane. I would accept it as a short communication and would cite the cleaned theorem if I needed the statement without the torsion-free caveat.

Referee Report

0 major / 6 minor

Summary. The paper removes the torsion-free hypothesis from Casalaina-Martin–Zhjeqi’s positivity theorem for logarithmic cotangent sheaves on smooth proper integral Deligne–Mumford stacks. The authors first prove Theorem A: if E is locally free on an integral DM stack and π: E ↠ Q is a coherent quotient, then the torsion subsheaf Q_tor is itself a quotient of a torsion-free coherent sheaf (namely the preimage E' = π^{-1}(Q_tor)). With c_1(Q_tor) thereby well-defined in the sense of their companion conventions, they deduce Theorem B: under the hypotheses of [CMZ26a, Thm. 4.3], every coherent quotient (not merely every torsion-free one) of every positive tensor power of Ω^1_X(log Δ) has pseudo-effective first Chern class. The argument uses the snake lemma, the fact that subsheaves of torsion-free sheaves are torsion-free, additivity of c_1, and the length formula for c_1 of torsion sheaves.

Significance. The note answers an explicit technical question raised by Casalaina-Martin–Zhjeqi in [CMZ26a, Rem. 4.5] and cleanly closes a small gap in the Chern-class calculus needed for their positivity theorem. The mathematics is elementary and standard once the [CMZ26b] conventions for c_1 of quotients of torsion-free sheaves are adopted; the contribution is modest in scope but correct and useful for anyone applying that theorem. Strengths include a self-contained proof of Theorem A from the snake lemma and the definition of torsion subsheaves, together with an explicit local DVR/Smith-normal-form verification of the length formula (Lemma 2.2). The result is a short, load-bearing clarification rather than a new positivity method.

minor comments (6)

[Title] In the running title/header the word “STACKS” is broken as “ST ACKS”; please correct the line break.
[§2.2] Section 2.2 and Lemma 2.2 rely on the Chern-class conventions of [CMZ26b, §1.5]. A one-sentence pointer that those conventions are exactly the ones used in [CMZ26a, Thm. 4.3] would make the match with the original theorem fully explicit for the reader.
[§3, Lemma 3.1] Lemma 3.1 is stated for noetherian DM stacks and proved by the snake lemma in the abelian category of coherent sheaves; a brief citation or remark that this abelianity is standard (e.g. Laumon–Moret-Bailly) would help non-specialists.
[Remark 1.4] Remark 1.4 on generative AI is unusually long for the introduction; consider shortening it and moving system details to an acknowledgement or footnote so the mathematical narrative is not interrupted.
[§4] In the proof of Theorem B the notation Q for the torsion-free quotient is easy to miss on a first reading; a more distinctive symbol (e.g. Q_tf) would improve clarity.
[References] References [CMZ26a] and [CMZ26b] are cited as arXiv preprints with 2026 identifiers; once final bibliographic data exist they should be updated.

Circularity Check

0 steps flagged

No significant circularity: Theorem A is elementary and independent; Theorem B only applies the external CMZ positivity theorem plus standard c1 additivity.

full rationale

The derivation chain is short and non-circular. Theorem A (Q_tor is a quotient of a torsion-free sheaf) is proved from the definition of the torsion subsheaf, the fact that subsheaves of torsion-free sheaves are torsion-free (Lemma 2.1), and the snake lemma on the kernel of E → Q → Q/Q_tor (Lemma 3.1); none of these steps define the conclusion in terms of itself or import it from prior work by Hou–Liu. Theorem B then takes the external Casalaina-Martin–Zhjeqi positivity theorem for the torsion-free quotient Q_bar, defines c1(Q_tor) via the paper’s stated conventions (which match the CMZ setup the theorem lives in), invokes the length formula of Lemma 2.2 to get effectivity of c1(Q_tor), and adds. The CMZ citations are to different authors and supply the hypothesis being relaxed, not a self-justifying uniqueness or ansatz. Mentions of Danus/Rethlas and Liu+26/Ju+26 concern how the note was produced and are not load-bearing for any mathematical step. No equation reduces the claim to a fitted quantity or to a definitional identity with the inputs. Score 0 is appropriate.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The note sits entirely on standard coherent-sheaf and intersection theory on DM stacks plus the prior positivity theorem it modifies. No free parameters or new entities are introduced. Load-bearing inputs are the original theorem, the Chern-class calculus for quotients of torsion-free sheaves, and basic properties of torsion subsheaves and DVRs at generic points of divisors.

axioms (6)

domain assumption Casalaina-Martin–Zhjeqi positivity theorem: under the stated geometric hypotheses, c₁ of every torsion-free coherent quotient of every positive tensor power of Ω¹_X(log Δ) is pseudo-effective ([CMZ26a, Thm 4.3]).
Invoked verbatim as Theorem 1.1; Theorem B is obtained by applying it to the torsion-free quotient Q/Q_tor.
domain assumption On a smooth integral DM stack, c₁ is defined for coherent sheaves that are quotients of torsion-free sheaves via det G' 儠 (det G'')^{-1}, is independent of presentation, and is additive in short exact sequences ([CMZ26b, §1.5], recalled in §2.2).
Without this calculus the phrase 'c₁(Q_tor) is well-defined and effective' has no meaning in the paper's language.
standard math The category of coherent sheaves on a noetherian DM stack is abelian; snake lemma applies to the diagram defining the preimage of a subsheaf.
Used in Lemma 3.1 to obtain surjectivity E' ↠ Q_tor.
standard math At the generic point of an integral codimension-one closed substack of a smooth integral DM stack, the local ring is a DVR; torsion lengths equal valuations of determinants via Smith normal form (Lemma 2.2).
Supplies the effectivity formula c₁(T) = ∑ length(T_η_D)[D] for torsion sheaves that are quotients of torsion-free sheaves.
standard math Subsheaves of torsion-free coherent sheaves are torsion-free (Lemma 2.1); locally free sheaves are torsion-free.
Used to conclude that the preimage E' is torsion-free.
domain assumption X is a smooth proper integral DM stack over C with projective coarse moduli space; Δ is a reduced normal-crossing divisor; some positive tensor power of the log cotangent contains a subsheaf with big determinant.
Geometric hypotheses copied from the original theorem; required for pseudo-effectivity on the coarse space.

pith-pipeline@v1.2.0-daily-grok45 · 9605 in / 3345 out tokens · 47449 ms · 2026-07-30T14:46:08.757464+00:00 · methodology

0 comments

read the original abstract

We remove the torsion-free hypothesis from a positivity theorem of Casalaina-Martin and Zhjeqi, answering a question of them. The main result of this paper was obtained using generative AI, particularly ChatGPT 5.5 Pro and the Danus system.

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 linked inside Pith

[1]

Casalaina-Martin and S

S. Casalaina-Martin and S. Zhjeqi, Foliations, slope stability, and positivity of log canonical bundles on Deligne--Mumford stacks, arXiv:2605.26443

Pith/arXiv arXiv
[2]

Casalaina-Martin and S

S. Casalaina-Martin and S. Zhjeqi, Movable curve classes and slope stability on Deligne--Mumford stacks, arXiv:2605.26101

Pith/arXiv arXiv
[3]

H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, S. Liu, L. Chen, Y. Wang, Y. Wang, Z. Wang, W. He, P. Wu, L. Xiao, R. Liu, B. Dai, and B. Dong, Automated Conjecture Resolution with Formal Verification, arXiv:2604.03789

Pith/arXiv arXiv
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J. Liu, G. Gao, Z. Sun, B. Wu, S. Liu, J. Jiang, H. Ju, L. Chen, R. Cheng, X. Zhang, and B. Dong, Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory, arXiv:2607.06447

Pith/arXiv arXiv

[1] [1]

Casalaina-Martin and S

S. Casalaina-Martin and S. Zhjeqi, Foliations, slope stability, and positivity of log canonical bundles on Deligne--Mumford stacks, arXiv:2605.26443

Pith/arXiv arXiv

[2] [2]

Casalaina-Martin and S

S. Casalaina-Martin and S. Zhjeqi, Movable curve classes and slope stability on Deligne--Mumford stacks, arXiv:2605.26101

Pith/arXiv arXiv

[3] [3]

H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, S. Liu, L. Chen, Y. Wang, Y. Wang, Z. Wang, W. He, P. Wu, L. Xiao, R. Liu, B. Dai, and B. Dong, Automated Conjecture Resolution with Formal Verification, arXiv:2604.03789

Pith/arXiv arXiv

[4] [4]

J. Liu, G. Gao, Z. Sun, B. Wu, S. Liu, J. Jiang, H. Ju, L. Chen, R. Cheng, X. Zhang, and B. Dong, Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory, arXiv:2607.06447

Pith/arXiv arXiv