$E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks

Haoxu Wang

REVIEW 2 major objections 4 minor 10 references

$E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The irregular Hodge spectral sequence degenerates at E1 on Deligne–Mumford stacks

desk verdict Theorem 1.1 is a genuinely new and sound finite-flat-descent proof of E1-degeneration for projective good stack compactifications; Theorem 1.2 is a conditional corollary that depends on the unproved comparison theorem in the companion paper [Wan26]. read the letter →

arxiv 2608.08539 v1 pith:JYCRXMVQ submitted 2026-08-09 math.AG

Haoxu Wang This is my paper · ORCID

classification math.AG MSC 14F4014A2014E0514J33

keywords irregularHodgefiltrationDeligne–MumfordstacksE1-degenerationfiniteflatdescentKontsevichcomplexestwisteddeRhamcohomologyspectralsequences

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

This paper proves that the irregular Hodge filtration on the twisted de Rham cohomology of a smooth separated Deligne–Mumford stack over $\mathbb{C}$ degenerates at the $E_1$ page of its spectral sequence, for every rational weight. The result holds on every good stack compactification and every NC rational stack compactification, so the filtration and its graded dimensions can be computed on whichever compactification is available. The proof reduces the stack case to the known Esnault–Sabbah–Yu theorem for smooth projective varieties by pulling a projective stack compactification back along a finite flat cover and descending degeneration through a filtered retract. A comparison theorem from the companion paper then transfers the statement from projective good models to arbitrary rational compactifications.

What carries the argument

The mechanism is finite flat descent for Kontsevich complexes. The paper constructs, for a projective good stack compactification $(X,D,\bar w)$, a finite flat surjective lci map $q:Y\to X$ from a smooth projective scheme such that $(q^{-1}D)_{\mathrm{red}}$ is SNC. The core identities are the weighted logarithmic pullback and trace maps $q^*: L^a_X(c)\to q_* L^a_Y(c)$ and $\mathrm{Tr}^a_q: q_* L^a_Y(c)\to L^a_X(c)$ on the rounded logarithmic lattices $L^a_X(c)=\Omega^a_X(\log D)(\lfloor cP\rfloor)$, with $\mathrm{Tr}_q\circ q^*=\deg(q)\,\mathrm{id}$. These maps restrict to the Kontsevich complexes, so normalized trace makes the filtered complex on $X$ a direct summand of that on $Y$; $E_1$-degeneration on $Y$, given by Esnault–Sabbah–Yu, then forces $E_1$-degeneration on $X$.

What would settle it

Compute the $E_1$ page and the differential $d_1$ for a projective good stack compactification with a stacky point and some rational weight $\alpha$; any nonzero $d_1$ would directly contradict Theorem 1.1, which predicts every such differential is zero.

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Extended reading notes

Core claim

The paper's central claim is that for any smooth separated Deligne–Mumford stack $\mathscr U$ over $\mathbb{C}$ with quasi-projective coarse space and any regular function $w$, the spectral sequence $E_1^{p,q}=H^q(X,\Omega^p_{X,\bar w}(\alpha))$ converging to $H^{p+q}(X,K_{X,\bar w}(\alpha))$ degenerates at $E_1$ for every $\alpha\in\mathbb{Q}\cap[0,1)$, on every projective good stack compactification and, via the companion comparison, on every NC rational stack compactification. In practical terms, the irregular Hodge filtration is strict: the cohomology of the total complex is the direct sum of the cohomologies of the individual graded pieces, and the graded dimensions are independent of the chosen compactification. The stack-level statement is the new part; the scheme case was established by Esnault–Sabbah–Yu.

Load-bearing premise

The argument depends on the companion paper's comparison theorem matching the spectral sequences from the $E_1$ page onward, since a match of only the total cohomology or final filtrations would not transfer degeneration to arbitrary rational compactifications.

Editorial extensions

If this is right

On every projective good stack compactification, the spectral sequence (1.1) degenerates at $E_1$, so $H^{p+q}(X,K_{X,\bar w}(\alpha))$ is the direct sum of the $H^q(X,\Omega^p_{X,\bar w}(\alpha))$.
On every NC rational stack compactification, both the fixed-$\alpha$ Kontsevich spectral sequence and the Yu integer-slice spectral sequences degenerate at $E_1$.
The irregular Hodge filtration and its graded dimensions can therefore be computed on any good or NC rational stack compactification, without finding a projective smooth model.
The theorem extends the Esnault–Sabbah–Yu degeneration from smooth projective varieties to smooth proper Deligne–Mumford stacks, including stacks whose coarse spaces are only quasi-projective.

Reading between the lines

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

The same pullback-and-trace descent could apply to other filtered complexes whose lattices satisfy the two inequalities $e\lfloor cm\rfloor\le \lfloor cem\rfloor$ and the trace calculation at codimension-one points; twisted Higgs and logarithmic de Rham complexes are natural candidates.
Because the rational case rests on the companion comparison identifying spectral sequences from $E_1$, the present paper's Theorem 1.2 is only as strong as that comparison; a reader relying on stack-level degeneration on arbitrary compactifications needs both papers.
One could test the theorem on explicit toric stack examples with stacky points at infinity, where the Kontsevich lattices and the spectral sequence can be computed by hand; the vanishing of all higher differentials would corroborate the stack-level statement.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The new thing is Theorem 1.1: E1-degeneration for projective good stack compactifications, proved by reducing to Esnault–Sabbah–Yu via a finite flat cover from a smooth projective scheme. The proof is internal and I think solid. The two technical cores are the cover construction (Theorem 3.3, via stratified Kresch–Vistoli plus relative Bertini) and the trace theorem on rounded logarithmic lattices (Theorem 4.6). I checked the local DVR calculation in Section 4.3 and it works; the pullback inequality is just the integer inequality floor(ce) ≥ e floor(c), and the trace containment follows from the Kummer trace formula. The descent argument in Theorem 5.1 is clean: a filtered retract inherits E1-degeneration page by page.

The soft spot is exactly what the stress test flags. Theorem 1.2, the statement for arbitrary NC rational compactifications, is not proved here. It is transferred from the companion paper's Theorem 2.5, which asserts that the derived-global-section isomorphisms identify the fixed-α Kontsevich spectral sequences from E1 onward. That is precisely the property needed; filtered quasi-isomorphisms on total complexes would not be enough. The present paper restates the theorem but does not prove it. So the NC rational case, and the “every good stack compactification” phrasing in the abstract, are conditional on [Wan26] being correct at that strength. A referee needs the companion in hand. Also the local freeness of the Kontsevich lattices and the existence of a projective good model come from [Wan26]. This is all disclosed, but it means the paper is not self-contained.

I don’t see circularity or fitting. The proof of Theorem 1.1 never assumes the degeneration on X; the retract argument goes the right way. The citation pattern is fine; the self-citation is to the companion that builds the compactification theory, and it is used for what it proves.

Who should read this: people working on irregular Hodge filtrations, twisted de Rham cohomology, or mirror symmetry on stacks. The finite flat descent technique for Kontsevich complexes is worth knowing. I would send it to a serious referee rather than desk reject. The referee should verify Theorem 2.5 against [Wan26] before the NC rational theorem is accepted; the projective stack theorem can be judged on its own.

Referee Report

2 major / 4 minor

Summary. The paper proves E1-degeneration of the Kontsevich spectral sequence for irregular Hodge filtrations on smooth separated Deligne--Mumford stacks over C with quasi-projective coarse space, at every rational index. Theorem 1.1 establishes this for projective good stack compactifications; Theorem 1.2 extends the statement to NC rational stack compactifications. The proof constructs a finite flat cover of the stack by a smooth projective scheme with SNC pullback boundary (Proposition 3.2 and Theorem 3.3), shows that pullback and trace preserve the rounded logarithmic lattices (Proposition 4.1 and Theorem 4.6), obtains a filtered retraction of the Kontsevich complexes (Proposition 4.7 and Corollary 4.8), and then transfers E1-degeneration from the Esnault--Sabbah--Yu theorem for smooth projective schemes (Theorem 5.1).

Significance. The finite-flat-descent method is a clean and apparently sound technique, and the proof of Theorem 1.1 is carefully executed. The local DVR calculation in Section 4.3, especially Lemma 4.5, is convincing. If the comparison statements from the companion paper [Wan26] are available, Theorems 1.1 and 1.2 give a complete E1-degeneration statement for irregular Hodge filtrations on Deligne--Mumford stacks, with direct applications to the stacky mirror-symmetry setting of Harder--Lee. The main caveat is that the paper's advertised Theorem 1.2 depends on an unproved comparison theorem quoted from the self-cited companion preprint [Wan26], so the paper is not self-contained for its full stated scope.

major comments (2)

[Section 5.2, proof of Theorem 1.2]
[Section 2.2 and Corollary 4.8] The manuscript also cites [Wan26, Proposition 3.5(i)] for the local freeness of the Kontsevich lattices Omega^a_{X,bar w}(alpha), and this local freeness is used in Corollary 4.8 to justify the q_*-acyclicity of the terms of the complex. In addition, Theorem 2.4, which provides the projective good compactification used in the proof of Theorem 1.2, is quoted from [Wan26, Proposition 5.4(ii)]. Thus even the proof of Theorem 1.1 depends on several unproved statements from the companion paper. These dependencies should be itemized clearly; if the companion is not yet published, the statements should be proved here or the relevant theorems marked as conditional.

minor comments (4)

[Eq. (2.3)] The display defining the Kontsevich lattice loses the quotient symbol in the codomain; writing Omega^{a+1}_X(*D) / L^{a+1}_X(alpha) explicitly would avoid ambiguity.
[Theorem 1.2] The five-point definition of an NC rational stack compactification is stated inside the theorem; moving it to a definition in Section 2 would improve readability.
[Lemma 3.1, part (2)] The sentence 'If dim B = 0, our choice of H makes this intersection empty' is terse; spelling out that a general hypersurface avoids a finite set of points would help.
[Theorem 4.6, proof] The assertion that checking the containment at every codimension-one point suffices would be clearer with one sentence explaining that the failure locus of a morphism into a locally free sheaf has pure codimension one unless the morphism is zero.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; Theorem 1.1 is proved internally and Theorem 1.2 depends on an independent companion comparison theorem, not on the conclusion.

full rationale

The paper's central new theorem (Theorem 1.1) is not circular. It constructs a finite flat surjective cover Y -> X from a smooth projective scheme (Theorem 3.3), proves that pullback and trace preserve the rounded logarithmic lattices (Proposition 4.1 and Theorem 4.6), obtains a filtered retraction (Proposition 4.7 and Corollary 4.8), and then descends E1-degeneration from the Esnault-Sabbah-Yu scheme theorem (Theorem 2.6) via Theorem 5.1. None of these steps assumes the conclusion. The only reliance on the author's companion preprint [Wan26] is through stated input theorems: local freeness of Kontsevich lattices, existence of a projective good compactification, and comparison of compactifications identifying the fixed-alpha Kontsevich spectral sequences from E1 onward (Theorem 2.5). Theorem 1.2 is genuinely conditional on the companion comparison theorem; if that comparison identified only abutments, Theorem 1.2 would not follow. But this is a dependency on an external (self-cited) theorem, not a circular reduction: the comparison theorem is not equivalent to E1-degeneration and does not have E1-degeneration among its hypotheses. No fitted parameter is renamed as a prediction, and no definition is made in terms of the target result. Thus the paper exhibits no circular step; the appropriate concern, if any, is whether [Wan26]'s comparison theorem is fully proved, which is a completeness or correctness issue rather than circularity.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The proof is a reduction to the external Esnault-Sabbah-Yu theorem, but it depends structurally on several results from the self-cited companion paper [Wan26]: local freeness of the Kontsevich lattices, existence of projective good compactifications, and comparison of compactifications identifying spectral sequences. No free parameters or invented entities appear. The main technical content of this paper, the finite flat cover and the trace theorem on rounded logarithmic lattices, is proved in full.

assumptions (8)

standard math Esnault-Sabbah-Yu theorem: E1-degeneration for smooth projective complex schemes (Theorem 2.6).
External theorem cited as [ESY17, Theorem 1.2.2 and Corollary 1.4.8]; the stack proof reduces to this via finite flat descent.
standard math Kresch's theorem: smooth separated DM stack with quasi-projective coarse space is a quotient stack.
Cited as [Kre09, Theorem 4.4] and used in Theorem 3.3 to start the Kresch-Vistoli cover construction.
standard math Kresch-Vistoli construction: existence of P, Q, S, U with the stated properties.
Cited as [KV04, Theorem 2.1]; the starting point of Proposition 3.2.
domain assumption Companion [Wan26, Proposition 3.5(i)]: Kontsevich lattices are locally free of the expected rank.
Self-cited companion result used in Section 2.2 and Corollary 4.8; not independently proved here.
domain assumption Companion [Wan26, Theorem 5.6]: comparison of compactifications identifies spectral sequences from E1.
Restated as Theorem 2.5; load-bearing for Theorem 1.2.
domain assumption Companion [Wan26, Proposition 5.4(ii)]: existence of a projective good stack compactification.
Restated as Theorem 2.4 and used in the proof of Theorem 1.2.
standard math Standard DVR facts: strict henselization decomposition and Kummer classification from Fu's book.
Used in Lemma 4.3; standard local field theory.
standard math Standard scheme facts: miracle flatness, finite iff proper plus quasi-finite, trace base change, and acyclicity of finite pushforwards.
Used in Proposition 3.2, Lemma 4.2, and Corollary 4.8 via Stacks project tags.

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Cite this review

Pith. "Pith review of $E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks." pith.science (2026). https://pith.science/paper/JYCRXMVQ

@misc{pith2026260808539,
  author       = {Pith},
  title        = {Pith review of: $E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JYCRXMVQ}},
  note         = {Machine review of arXiv:2608.08539}
}

abstract

Let $(\mathscr U,w)$ be a smooth separated Deligne--Mumford stack over $\mathbb{C}$. Assume that its coarse space is quasi-projective. We prove $E_1$-degeneration at every rational index for its irregular Hodge filtration. The result holds on every good stack compactification and every NC rational stack compactification. In particular, the filtration and its graded dimensions can be computed on any such compactification. The proof uses finite flat descent for Kontsevich complexes. It reduces the projective stack case to the theorem of Esnault--Sabbah--Yu for smooth projective varieties.

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Reference graph

Works this paper leans on

10 extracted references · 5 canonical work pages

[1]

Methods Appl

Kai-Chieh Chen and Jeng-Daw Yu, The K\"unneth formula for the twisted de Rham and Higgs cohomologies , SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 055, 14

2018
[2]

Reine Angew

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[3]

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arXiv 2025
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London Math

Andrew Kresch and Angelo Vistoli, On coverings of Deligne--Mumford stacks and surjectivity of the Brauer map , Bull. London Math. Soc. 36 (2004), no. 2, 188--192

work page 2004
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The Stacks Project Authors , Stacks project, https://stacks.math.columbia.edu, 2026

2026

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[9]

Haoxu Wang, Compactification independence of the irregular Hodge filtration on Deligne--Mumford stacks , 2026, https://arxiv.org/abs/2608.06234 arXiv:2608.06234

2026 arXiv
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Jeng-Daw Yu, Irregular Hodge filtration on twisted de Rham cohomology , Manuscripta Math. 144 (2014), 99--133

2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Methods Appl

Kai-Chieh Chen and Jeng-Daw Yu, The K\"unneth formula for the twisted de Rham and Higgs cohomologies , SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 055, 14

2018

[2] [2]

Reine Angew

H \'e l \`e ne Esnault, Claude Sabbah, and Jeng-Daw Yu, E_1 -degeneration of the irregular Hodge filtration , J. Reine Angew. Math. 729 (2017), 171--227

work page 2017

[3] [3]

13, World Scientific, Hackensack, NJ, 2011

Lei Fu, \'E tale cohomology theory , Nankai Tracts in Mathematics, vol. 13, World Scientific, Hackensack, NJ, 2011

work page 2011

[4] [4]

52, Springer-Verlag, New York--Heidelberg, 1977

Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, New York--Heidelberg, 1977

work page 1977

[5] [5]

Andrew Harder and Sukjoo Lee, Irregular Hodge numbers of stacky Clarke mirror pairs , 2025, arXiv:2408.09016v2

arXiv 2025

[6] [6]

Part 1, Proc

Andrew Kresch, On the geometry of Deligne--Mumford stacks , Algebraic Geometry---Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, American Mathematical Society, Providence, RI, 2009, pp. 259--271

2005

[7] [7]

London Math

Andrew Kresch and Angelo Vistoli, On coverings of Deligne--Mumford stacks and surjectivity of the Brauer map , Bull. London Math. Soc. 36 (2004), no. 2, 188--192

work page 2004

[8] [8]

The Stacks Project Authors , Stacks project, https://stacks.math.columbia.edu, 2026

2026