Compactification Independence of the Irregular Hodge Filtration on Deligne-Mumford Stacks

Haoxu Wang

REVIEW 3 major objections 3 minor 1 cited by

Compactification Independence of the Irregular Hodge Filtration on Deligne-Mumford Stacks

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The irregular Hodge filtration on a smooth Deligne–Mumford Landau–Ginzburg model is independent of the chosen compactification, making the orbifold irregular Hodge numbers invariants of the pair.

desk verdict The local root and blowup comparisons are solid and new; the global independence result is a well-argued reduction to an unverified Bergh–Rydh theorem, so referee it but make the dependency explicit. read the letter →

arxiv 2608.06234 v1 pith:AOOOUTUV submitted 2026-08-06 math.AG

Haoxu Wang This is my paper · ORCID

classification math.AG MSC 14F4014A2014E0514J33

keywords irregularHodgefiltrationDeligne–MumfordstacksLandau–GinzburgmodelscompactificationindependencetwisteddeRhamcohomologyweakfactorizationKontsevichlatticesorbifoldnumbers

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 attached to a smooth Deligne–Mumford Landau–Ginzburg model $(U,w)$ is independent of the compactification used to define it. For smooth varieties this was Yu's theorem; the paper extends it to Deligne–Mumford stacks by allowing compactifications whose boundary is normal crossing and whose potential extends only rationally, subject to a local nondegeneracy condition near the polar divisor. The main result states that any two such rational stack compactifications produce the same filtered subspaces $F^\lambda_{\mathrm{irr}}H^k(U,w)$ of exponentially twisted de Rham cohomology. Because the inertia stack sectors inherit the same property, the orbifold irregular Hodge numbers of Harder and Lee become invariants of the pair alone, which is exactly what their mirror-symmetry formula requires.

What carries the argument

The carrying object is the Yu filtered complex $F^\lambda_{\mathrm{Yu}}K^a_{X,w} = \Omega^a_X(\log D)(\lfloor(a-\lambda)P\rfloor)$ for $a \geq \lceil\lambda\rceil$ and zero otherwise, whose hypercohomology image defines the filtration. To compare two compactifications the paper uses the Kontsevich lattices $\Omega^a_{X,w}(\alpha) = \ker(\nabla_w)$ inside the meromorphic de Rham complex; these lattices pull back functorially even when rounded Yu lattices do not, and the comparison is mediated by roofs through them. The geometric input is a filtered comparison for two elementary modifications — root constructions along boundary divisors and ordinary blowups of boundary-admissible centers — verified through an acyclic-defect criterion (vanishing of $R\pi_*$ of the cokernels). Global comparison then follows from two structural results: every rational compactification admits a good resolution by boundary-admissible blowups after toroidal strictification, and any two good compactifications are joined by a zigzag of such blowups and roots via relative stacky weak factorization.

What would settle it

Take a smooth proper Deligne–Mumford curve with two boundary components and a rational function $w$ with poles on both, form two compactifications by rooting different boundary components, and compute the level-$\lambda$ Yu images in $H^1(U,w)$; any discrepancy between the two subspaces, or any pair of good compactifications not joinable by boundary blowups and roots, would refute Theorem 5.6.

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Extended reading notes

Core claim

The paper establishes compactification independence for the Yu irregular Hodge filtration in the setting of smooth separated Deligne–Mumford stacks of finite type over $\mathbb{C}$. Given a regular function $w$, twisted de Rham cohomology $H^k(U,w)$ carries, for each rational $\lambda$, a subspace $F^\lambda_{\mathrm{irr},X}H^k$ defined as the image of the hypercohomology of Yu's filtered complex $F^\lambda_{\mathrm{Yu}}K^\bullet_{X,w}$. Theorem 5.6(iv) asserts that for any two rational stack compactifications $X_1$ and $X_2$ these subspaces are literally equal inside $H^k(U,w)$, for all $k$ and $\lambda$; consequently the orbifold irregular Hodge numbers $f^{\lambda,\mu}_{\mathrm{orb}}(U,w)$ of Harder–Lee depend only on $(U,w)$ and not on any chosen compactification.

Load-bearing premise

The global comparison in Theorem 5.6 rests on the relative stacky weak factorization theorem of Bergh–Rydh, cited from an arXiv preprint and not proved or verified in the paper, together with Harper's strictification and Rydh's compactification results; if any of these unpublished results fail in the needed generality, the zigzag connecting two good compactifications may not exist and the theorem would not follow.

Editorial extensions

If this is right

For every choice of compactifications of the inertia sectors, the age-shifted orbifold filtration $F^\lambda_{\mathrm{orb}}H^q_{\mathrm{dR,orb}}(U,w)$ is independent of the choice, so the Harder–Lee orbifold irregular Hodge numbers $f^{\lambda,\mu}_{\mathrm{orb}}(U,w)$ are invariants of the pair $(U,w)$ alone.
The stacky Clarke mirror formula of Harder–Lee becomes a statement about invariants rather than about chosen compactifications, since both sides are now well-defined without auxiliary data.
Any two compactifications produce filtered spectral sequences that agree from the $E_1$ page onward; in particular, degeneration at $E_1$ is a property of the Landau–Ginzburg model, not of the compactification.
Good stack compactifications exist for every smooth separated Deligne–Mumford Landau–Ginzburg model, and projective good compactifications exist whenever the coarse moduli space is quasi-projective, so the invariants are computable in practice.

Reading between the lines

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

A natural testable extension, not pursued in the paper, is whether compactification independence persists in smooth families of Landau–Ginzburg models; if it does, the orbifold irregular Hodge numbers would vary flatly and could be computed at any special fibre.
The acyclic-defect criterion isolates the exact vanishing that makes each elementary modification invisible to the filtration; the same criterion could be reused to test other modifications, such as weighted or non-smooth toroidal blowups, whenever the corresponding defect vanishing can be verified.
Because the theorem works with rational extensions of $w$ and permits zero–pole intersections satisfying the Chen–Yu normal form, computational recipes for irregular Hodge numbers can now be run on any convenient normal-crossing compactification, including ones where the potential is not a morphism.
The sectorwise application suggests that orbifold irregular Hodge numbers could be studied one inertia component at a time, keeping only the age shift and the restricted potential; the paper does not explore whether such sectorwise invariants satisfy further structural identities such as symmetries under crepant resolutions.

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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

Read it. The kernel of the paper is the local comparison machinery: Proposition 3.13 reduces filtered invariance to vanishing of Kontsevich-lattice defects, and then Theorem 4.4 (root stacks) and Theorem 4.11 (boundary-admissible blowups) verify that criterion. The root comparison is genuinely new and essentially self-contained. The blowup comparison is plausible and covers the zero–pole centers needed later. That part is worth taking seriously.

The global theorem is where the paper leans on outside scaffold. Theorem 5.5, which connects any two good compactifications by a zigzag of boundary blowups and roots, is a direct citation to Bergh–Rydh [BR19, Theorem D], an unpublished arXiv preprint. The stress-test note is right: the text never checks that the theorem applies to arbitrary good stack compactifications, particularly when coarse spaces are non-projective, nor that the relative-to-U zigzag preserves the potential. If BR19 is invalid or inapplicable, Theorem 5.6(iv) falls. This is load-bearing, not a footnote. Harper [Har17] and Rydh [Ryd11] are also unpublished, which matters for Theorem 5.3 and Proposition 5.4, though less dramatically since a published alternative may exist.

Two soft spots beyond the external dependencies. The module calculation in Theorem 4.11 — the five-term filtration of the Kontsevich cokernel — is summarized rather than written out; a referee will have to work through it. And the paper does not state the main theorem as conditional on BR19, which is the honest way to present it while BR19 remains a preprint.

Who is this for? People in irregular Hodge theory and orbifold mirror symmetry wanting the stack extension of Yu and Chen–Yu. The orbifold corollary for Harder–Lee numbers is useful. I would send it to a serious referee. The local theorems are solid and publishable; the global claim needs either a verification of BR19's hypotheses or an explicit conditionality. Referee time is justified.

Referee Report

3 major / 3 minor

Summary. The paper studies the irregular Hodge filtration on the twisted de Rham cohomology of a smooth separated Deligne-Mumford Landau-Ginzburg model (U,w). It introduces NC rational stack compactifications, in which the potential w extends only rationally and is required to satisfy a local nondegeneracy condition near its polar divisor. It defines the Yu-type filtered complex and the Kontsevich lattices on such compactifications, and the main theorem (Theorem 5.6) asserts that the resulting filtration on H^k(U,w) is independent of the compactification. The proof reduces the comparison to an acyclic-defect criterion (Proposition 3.13), establishes filtered comparison for boundary roots (Theorem 4.4) and for boundary-admissible blowups (Theorem 4.11), resolves every NC rational compactification to a good stack compactification (Theorem 5.3), and finally invokes a relative stacky weak factorization (Theorem 5.5) to connect any two good compactifications. Corollary 5.7 transfers this independence to the orbifold irregular Hodge numbers of Harder and Lee via inertia sectors and age shifts.

Significance. If the main theorem is valid, it extends Yu's and Chen-Yu's compactification independence from smooth varieties to Deligne-Mumford stacks and makes the Harder-Lee orbifold irregular Hodge numbers invariants of the Landau-Ginzburg model, which is a meaningful step for stacky mirror symmetry. The paper's own local contributions are substantial: the roof construction via Kontsevich lattices cleanly handles zero-pole cancellation, the acyclic-defect criterion is a useful organizing principle, and the root and blowup comparisons are carried out with explicit defect filtrations and appear internally coherent. There is no circularity in the argument: independence is obtained from explicit quasi-isomorphisms, with external results used as tools. However, the global conclusion currently rests on substantial unpublished external inputs, most importantly the Bergh-Rydh weak factorization theorem, whose hypotheses are not verified in the manuscript.

major comments (3)

[§5.4, Theorem 5.5] The global zigzag connecting two arbitrary good stack compactifications is asserted by direct citation to the unpublished arXiv preprint [BR19, Theorem D], with the paper's own proof consisting of a short paragraph that does not state the hypotheses of that theorem. This is load-bearing: Theorem 5.5 is the only bridge between two arbitrary good models, and Theorem 5.6(iv) inherits all of its risk. In particular, Definition 2.3 only requires properness, and the paper explicitly notes that the coarse moduli space need not be projective, while many available weak factorization statements are formulated under projectivity or global-quotient hypotheses. The published global-quotient case [Ber18] does not cover the stated generality. The authors should quote [BR19, Theorem D] with its full hypotheses and verify them for good stack compactifications, provide a proof of Theorem 5.5, or restrict the main theorem to a class for which the factorization is established.
[§5.2–§5.3, Theorems 5.3 and 5.4] The reduction to good compactifications depends on Harper's unpublished strictification [Har17, Lemmas A.2.4 and A.2.6] and, for the existence statement, on Rydh's unpublished draft [Ryd11, Theorem F]. These are not bibliographic details: Proposition 5.2 and Theorem 5.3 need the specific centers produced by Harper's cone-complex construction to be boundary strata preserving the polar normal form, and Proposition 5.4 needs the full force of Rydh's compactification theorem. The manuscript gives no independent verification of these inputs. If these results are not available in a stable, verifiable form, the resolution theorem and the sectorwise existence used in Corollary 5.7 are not self-contained. The authors should make the precise statements of the cited results available in an appendix or replace them with proofs.
[§3.5, Theorem 3.17] The advertised application to Harder-Lee orbifold irregular Hodge numbers depends on the identification of their sector complexes with the Yu complexes. The proof of Theorem 3.17 asserts that Harder-Lee's explicit Yu-complex formula is 'termwise the complex of Definition 3.2' and appeals to Lemma 3.16(ii), but the passage from Harder-Lee's nondegeneracy conditions, which include a global irreducibility requirement on the zero divisor, to the paper's weaker NC rational condition is only explained in an informal paragraph before Definition 3.15. Since the restriction of the zero divisor to an inertia sector can split or become empty, this is a real convention rather than an immediate consequence. The identification in Theorem 3.17 should be proved in detail, or explicitly stated as an additional assumption on the Harder-Lee compactifications.

minor comments (3)

[Throughout] The text contains many encoding artifacts and typos (for example 'èsimultaneous', 'ètale', 'trange') that should be cleaned before publication.
[§2.2, Definition 2.3] The terminology 'NC rational', 'strict', 'morphic', and 'good' is introduced in quick succession; a short glossary or summary table of the four levels would improve readability, especially since several theorems mix these adjectives.
[References] The references [BR19], [Har17], and [Ryd11] are listed as unpublished preprints or preliminary drafts; please update to published versions if they exist, otherwise state their status and provide stable identifiers.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the compactification independence theorem is proved by explicit filtered quasi-isomorphisms for elementary modifications, and the global zigzag is imported as an external geometric factorization result, not assumed from the target conclusion.

full rationale

The paper's derivation chain is: define the Yu filtration on any NC rational stack compactification; prove filtered comparison under the two elementary modifications (boundary roots and boundary-admissible blowups) via explicit Kontsevich-lattice defect computations (Theorems 4.4 and 4.11); resolve an arbitrary NC rational compactification to a good one by strictification and zero–pole blowups (Theorem 5.3); join any two good compactifications by a zigzag supplied by Bergh–Rydh relative stacky weak factorization (Theorem 5.5); and compose the comparisons to obtain equality of image subspaces (Theorem 5.6). No step defines the desired independence into the filtration or into a fitted parameter. In particular, Theorem 5.5 is presented as an external geometric input: the paper explicitly cites [BR19, Theorem D] and notes the published global-quotient case [Ber18]. Even if this external theorem is heavy or its hypotheses are not fully checked, relying on an independent geometric statement is dependency, not circularity. The local comparisons are also not assumed: the acyclic-defect criterion (Proposition 3.13) is verified by computing and killing exceptional cokernels, and Corollary 3.11 converts the resulting quasi-isomorphisms into equality of cohomological images. There is no fitted input later relabeled as a prediction, no self-citation load-bearing for the central claim, and no renaming of a known empirical pattern as an organizing principle. The paper is not fully self-contained because it leans on unpublished or preprint results, but that is a correctness-risk issue, not circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper introduces no free parameters or new physical or mathematical entities. The central claim is built on a set of substantial external results: Deligne's logarithmic comparison, Yu's filtration theory, the Chen-Yu local Kontsevich comparison, Cadman's root stacks, AOV tame pushforward, and, at the global level, the stack-theoretic results of Harper, Bergh-Rydh, Rydh, Temkin, and Kresch. The first group are foundational and standard; the last group includes unpublished or preliminary preprints whose correctness is not verified in this paper.

assumptions (9)

standard math Deligne's logarithmic comparison theorem for NC boundaries and connections with zero residues.
Identifies the log de Rham complex with the derived direct image of the twisted de Rham complex on the open model; used in Proposition 3.1.
domain assumption Chen-Yu local Kontsevich-Yu comparison ([CY18, Proposition 1(ii)]).
Proposition 3.5(ii) relies on this published result to establish the filtered quasi-isomorphism between Kontsevich lattices and Yu complexes.
standard math Cadman's root stack description (quotient chart, properness, complement behavior).
Used in Proposition 4.2 to describe the local structure of root stacks along boundary divisors.
standard math AOV tame pushforward exactness for finite linearly reductive stabilizers.
Used in Proposition 4.2 and Theorem 4.4 to show exactness of direct images for root morphisms and coarse moduli maps.
domain assumption Harper's toroidal boundary strictification ([Har17, Lemmas A.2.4 and A.2.6]).
Used in Proposition 5.2 and Theorem 5.3 to convert NC boundaries into finite labelled SNC boundaries by smooth star subdivisions.
domain assumption Bergh-Rydh relative stacky weak factorization ([BR19, Theorem D]).
Used in Theorem 5.5 to connect any two good stack compactifications via a zigzag of blowups and roots; the reference is an unrefereed arXiv preprint.
domain assumption Rydh's compactification theorem for tame Deligne-Mumford stacks ([Ryd11, Theorem F]).
Used in Proposition 5.4 to produce a proper ambient stack containing the given LG model; the reference is a preliminary draft.
domain assumption Temkin's functorial desingularization with boundary for algebraic stacks ([Tem18, Theorems 1.1.11 and 1.1.13(i)]).
Used in Proposition 5.4 to desingularize the graph closure and make the boundary a union of coordinate hyperplanes.
domain assumption Kresch's quotient and projective compactification results for Deligne-Mumford stacks ([Kre09, Theorems 4.4 and 5.3]).
Used in Proposition 5.4 to obtain a projective ambient stack when the coarse moduli space of U is quasi-projective.

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Pith. "Pith review of Compactification Independence of the Irregular Hodge Filtration on Deligne-Mumford Stacks." pith.science (2026). https://pith.science/paper/AOOOUTUV

@misc{pith2026260806234,
  author       = {Pith},
  title        = {Pith review of: Compactification Independence of the Irregular Hodge Filtration on Deligne-Mumford Stacks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AOOOUTUV}},
  note         = {Machine review of arXiv:2608.06234}
}

abstract

Let $(\mathscr U,w)$ be a smooth separated Deligne--Mumford stack of finite type over $\mathbb C$, with a regular function $w$. Following Yu, one can use a compactification to define a filtration on its twisted de Rham cohomology. The same compactification gives Kontsevich lattices that compute this filtration. The extension of $w$ on the compactification may be only rational. Only a local nondegeneracy condition near the polar divisor is imposed. We prove that the resulting filtration does not depend on the compactification. The proof uses good resolutions and stacky weak factorization. This reduces the comparison to blowups and roots along boundary divisors. Filtered comparison theorems for the Yu and Kontsevich complexes are established for both operations. Harder and Lee use orbifold irregular Hodge numbers in their study of stacky Clarke mirror pairs. Compactification independence applies sector by sector to their setting. Thus the resulting orbifold filtration and irregular Hodge numbers depend only on the stack Landau--Ginzburg model.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

$E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks
math.AG 2026-08 accept novelty 7.0 of 10

For every smooth separated Deligne-Mumford stack with quasi-projective coarse space, the irregular Hodge filtration degenerates at E1 on every good stack compactification.

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Works this paper leans on

23 extracted references · 22 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-- W itten theory of Deligne--Mumford stacks , Amer. J. Math. 130 (2008), no. 5, 1337--1398

work page 2008

[2] [2]

Dan Abramovich, Kalle Karu, Kenji Matsuki, and Jaros aw W odarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), 531--572

work page 2002

[3] [3]

Dan Abramovich, Martin Olsson, and Angelo Vistoli, Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008), 1057--1091

work page 2008

[4] [4]

Borisov, Linda Chen, and Gregory G

Lev A. Borisov, Linda Chen, and Gregory G. Smith, The orbifold Chow ring of toric Deligne--Mumford stacks , J. Amer. Math. Soc. 18 (2005), 193--215

work page 2005

[5] [5]

Daniel Bergh, Weak factorization and the Grothendieck group of Deligne--Mumford stacks , Adv. Math. 340 (2018), 193--210

work page 2018

[6] [6]

Niels Borne and Amine Laaroussi, Parabolic connections and stack of roots, Bull. Sci. Math. 187 (2023), Paper No. 103294, 33

work page 2023

[7] [7]

Daniel Bergh and David Rydh, Functorial destackification and weak factorization of orbifolds, 2019, arXiv:1905.00872v1

arXiv 2019

[8] [8]

Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), 405--427

work page 2007