{"id":"f88cf772-9892-4fbd-a1b4-faba7f088252","arxiv_id":"2608.06234","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Compactification independence of the irregular Hodge filtration is proven for Deligne-Mumford stacks, and the resulting orbifold irregular Hodge numbers become invariants of the stacky Landau-Ginzburg model.","lead":"Pure-mathematics paper proving that the irregular Hodge filtration on a Deligne-Mumford stack with a regular function is independent of the compactification used to define it. This makes certain orbifold mirror-symmetry invariants (Harder-Lee irregular Hodge numbers) well-defined properties of the Landau-Ginzburg model rather than of auxiliary choices.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Global independence rests on the unverified Bergh–Rydh weak factorization [BR19, Theorem D], whose hypotheses against non-projective coarse spaces and the relative-to-U zigzag are never checked.","rationale":"The paper's local analysis is extensive: root modifications (Theorem 4.4) and boundary-admissible blowups (Theorem 4.11) are treated with explicit filtered comparisons, and the passage from local quasi-isomorphisms to equality of image subspaces (Corollary 3.11) is logically clean. The genuinely load-bearing point is global: to compare two arbitrary NC rational compactifications, the proof first resolves them to good models and then joins the good models by the relative stacky weak factorization theorem of Bergh–Rydh. This is the single point where the entire zigzag, and therefore the claimed independence, is outsourced to an unpublished and not independently verified result. The paper even emphasizes that the intermediate coarse spaces need not be projective, which makes it particularly important to confirm that [BR19, Theorem D] applies in that generality. This is a correctness risk rather than a disagreement with mathematical consensus, and it matches the reader's identified weakest assumption. The proposed check is therefore to verify the theorem's hypotheses in the exact relative setting used in Theorem 5.5; if the check fails, the main theorem must be restricted, and if it passes, the conditional verdict can be upgraded.","tokens_in":32051,"tokens_out":30338,"duration_ms":330865,"concrete_test":"Extract the exact statement of [BR19, Theorem D] and check three hypotheses against Theorem 5.5: (1) the result must apply to smooth proper DM stacks with labelled SNC boundary whose coarse moduli spaces are proper algebraic spaces, not only projective schemes; (2) it must yield a zigzag with every edge the identity over the fixed open U; (3) every intermediate must admit compatible morphisms to P^1 extending w. If any hypothesis is absent, attempt the construction in a minimal example of two good compactifications with non-projective coarse spaces; failure of the zigzag would disprove Theorem 5.6, while success would resolve the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem depends on Theorem 5.5, whose proof is a direct citation to Bergh–Rydh [BR19, Theorem D], an unpublished arXiv preprint, together with Harper [Har17] and Rydh [Ryd11]. The paper asserts without checking that this theorem provides a relative zigzag with all arrows the identity over U and with compatible extensions of w. The concrete risk is that [BR19, Theorem D] may have hypotheses not satisfied by arbitrary good stack compactifications: the paper explicitly notes that coarse spaces need not be projective, while available statements of destackification/weak factorization are often phrased for projective coarse moduli spaces or for a single birational morphism rather than a relative pair of compactifications. If the theorem is invalid or inapplicable in this generality, the zigzag joining the two good resolutions does not exist, and Theorem 5.6(iv) is unsupported. The local filtered comparisons appear coherent, and no internal contradiction is evident, but the global compactification independence claim is not self-contained and inherits all risk from this external input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":32118,"tokens_out":28592,"duration_ms":289588,"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":[{"comment":"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.","section":"§5.4, Theorem 5.5"},{"comment":"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.","section":"§5.2–§5.3, Theorems 5.3 and 5.4"},{"comment":"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.","section":"§3.5, Theorem 3.17"}],"minor_comments":[{"comment":"The text contains many encoding artifacts and typos (for example 'èsimultaneous', 'ètale', 'trange') that should be cleaned before publication.","section":"Throughout"},{"comment":"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.","section":"§2.2, Definition 2.3"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is conditional on unpublished work by Bergh-Rydh, Harper, and Rydh, and the most important of these, [BR19, Theorem D], is cited without a statement of its hypotheses. The local comparison proofs are careful and appear sound, so the paper is likely salvageable, but the editor should require the factorization input to be either verified in detail or supplied as a proof. If the journal's policy discourages reliance on unpublished preprints for main theorems, this may also be a scope concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nTwo 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.\n\nWho 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.","headline":"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.","tokens_in":32746,"tokens_out":3108,"would_cite":true,"duration_ms":29399,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F40","14A20","14E05","14J33"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["irregular Hodge filtration","Deligne–Mumford stacks","Landau–Ginzburg models","compactification independence","twisted de Rham cohomology","weak factorization","Kontsevich lattices","orbifold irregular Hodge numbers"],"falsifier":"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.","tokens_in":31734,"feed_emoji":"🪞","tokens_out":8898,"duration_ms":74975,"temperature":0.7,"pith_summary":"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.","feed_headline":"Compactification no longer matters for irregular Hodge filtration","feed_subtitle":"Any two compactifications give the same twisted de Rham filtration, so orbifold Hodge numbers are invariants of the pair.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the irregular Hodge filtration and its compactification-independence proof for smooth varieties, the template this paper extends to stacks.","marker":"[Yu14]"},{"why":"Defines nondegenerate rational compactifications and the zero–pole blowup comparison used in the resolution step.","marker":"[CY18]"},{"why":"Supplies the Kontsevich-complex comparison that underlies the local Kontsevich–Yu quasi-isomorphism.","marker":"[ESY17]"},{"why":"Provides Theorem D, the relative stacky weak factorization by blowups and roots that connects any two good compactifications.","marker":"[BR19]"},{"why":"Supplies toroidal boundary strictification and the stacky factorization lemmas needed to resolve rational models.","marker":"[Har17]"},{"why":"Gives the tame Deligne–Mumford compactification theorem used to establish existence of good compactifications.","marker":"[Ryd11]"},{"why":"Supplies functorial desingularization with boundary for algebraic stacks in the existence proof.","marker":"[Tem18]"},{"why":"Defines orbifold irregular Hodge numbers and the mirror-pair context that motivates the sectorwise application.","marker":"[HL25]"},{"why":"Gives the quotient description of root stacks used in the boundary-root filtered comparison.","marker":"[Cad07]"},{"why":"Provides exactness of tame pushforward, used to kill the exceptional defects for root modifications.","marker":"[AOV08]"}],"fun_headline_variants":["Compactification no longer affects irregular Hodge filtration","Irregular Hodge filtration independent of compactification","Orbifold Hodge numbers now compactification-free","Same irregular Hodge filtration across compactifications","Stacky irregular Hodge filtration ignores compactification"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Compactification no longer affects irregular Hodge filtration","Irregular Hodge filtration independent of compactification","Orbifold Hodge numbers now compactification-free","Same irregular Hodge filtration across compactifications","Stacky irregular Hodge filtration ignores compactification"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000955,"raw_usage":{"total_tokens":4067,"prompt_tokens":938,"completion_tokens":3129,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":3057}},"tokens_in":554,"tokens_out":3129,"duration_ms":19743,"temperature":1.0,"reasoning_tokens":3057,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:52:48.897988+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}