{"id":"57adec6c-0a00-4fdb-9880-802a791eddf5","arxiv_id":"2608.08539","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every smooth separated Deligne-Mumford stack with quasi-projective coarse space, the irregular Hodge filtration degenerates at E1 on every good stack compactification.","lead":"This paper proves that the irregular Hodge filtration on a Deligne-Mumford stack with a regular function degenerates at the first page, as it does for smooth projective varieties. The proof constructs a finite flat cover by a smooth projective scheme and uses a trace map to transfer the known scheme result to the stack.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 hinges on the unproved comparison theorem from the companion paper [Wan26]; the paper only proves Theorem 1.1 internally, so the NC rational case is conditional on that external result.","rationale":"The reader's weakest assumption identifies exactly this concern, and I agree with that identification. The rest of the paper is detailed and I found no internal inconsistency: the trace calculation at codimension-one points, the strict-henselization reduction, and the descent argument for degeneration are coherent. The single load-bearing issue is external dependence on a self-cited companion preprint. Acceptance should be conditional on verification of the E1-level comparison stated in Theorem 2.5. If that comparison is valid, Theorem 1.2 follows and the paper's claims stand. If it is not, the rational-compactification part of the abstract fails, even though Theorem 1.1 remains valid. Thus I recommend CONDITIONAL acceptance rather than unconditional ACCEPT.","tokens_in":10958,"tokens_out":14909,"duration_ms":161087,"concrete_test":"Open [Wan26, Proposition 3.5 and Theorem 5.6] and check whether the comparison is constructed as a filtered quasi-isomorphism on the filtered complexes σ≥•K(α), with explicit isomorphisms on every filtration level and every successive quotient, from which the E1-page isomorphism and d_r=0 follow. Equivalently, re-derive the comparison in the special case where one compactification is a projective good model and the other is obtained by a boundary blow-up; compute H^q(X, Ω^p_{X,\\bar w}(α)) on both sides and verify the induced E1-isomorphism is compatible with differentials. If the companion only compares total cohomology or the abutment filtration, the transfer in Theorem 1.2 is unjustified.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The finite-flat-descent proof is internal and appears sound: Proposition 3.2 constructs a finite flat cover by a smooth projective scheme, Theorem 4.6 establishes the weighted logarithmic trace, and Theorem 5.1 transfers E1-degeneration. This is enough for Theorem 1.1. The extension to every NC rational stack compactification, Theorem 1.2, is then delegated to Theorem 2.5, restated from the self-cited companion [Wan26]. That theorem asserts that for two NC rational compactifications the compatible isomorphisms of derived global sections identify the fixed-α Kontsevich spectral sequences from E1 onward. This is exactly the property required: filtered quasi-isomorphisms on total complexes would only identify abutments; one needs the E1 pages and all subsequent differentials to match. The present paper does not prove this comparison or reduce it to a known theorem. If the companion comparison only identifies total cohomology or the induced filtration, Theorem 1.2 does not follow. Since [Wan26] is a separate preprint rather than a published theorem, the paper's central claim is not self-contained at this point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":11113,"tokens_out":15876,"duration_ms":168653,"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":[{"comment":"","section":"Section 5.2, proof of Theorem 1.2"},{"comment":"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.","section":"Section 2.2 and Corollary 4.8"}],"minor_comments":[{"comment":"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.","section":"Eq. (2.3)"},{"comment":"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.","section":"Theorem 1.2"},{"comment":"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.","section":"Lemma 3.1, part (2)"},{"comment":"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.","section":"Theorem 4.6, proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically well written and the internal proof of Theorem 1.1 appears sound. However, the advertised Theorem 1.2 and several structural inputs depend on the author's own companion preprint [Wan26]. If the companion is not yet accepted or is unavailable to the referees, the present manuscript cannot be checked for the full scope of its claims. You may wish to verify that [Wan26] is publicly available and that its statements are sufficient before final acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nI 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.\n\nWho 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.","headline":"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].","tokens_in":11700,"tokens_out":11864,"would_cite":true,"duration_ms":122571,"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 spectral sequence degenerates at E1 on Deligne–Mumford stacks","keywords":["irregular Hodge filtration","Deligne–Mumford stacks","E1-degeneration","finite flat descent","Kontsevich complexes","twisted de Rham cohomology","spectral sequences"],"falsifier":"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.","tokens_in":10692,"feed_emoji":"📐","tokens_out":10235,"duration_ms":92042,"temperature":0.7,"pith_summary":"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.","feed_headline":"Stacks get E1 degeneration for irregular Hodge filtrations","feed_subtitle":"A finite flat descent trick transfers Esnault–Sabbah–Yu from projective varieties to every good stack compactification.","key_machinery":"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 $X$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Companion paper that supplies the comparison theorem identifying spectral sequences from E1, the existence of projective good models, and the local-freeness of Kontsevich lattices.","marker":"[Wan26]"},{"why":"Esnault–Sabbah–Yu theorem for smooth projective varieties, the base case the stack proof descends from.","marker":"[ESY17]"},{"why":"Kresch–Vistoli stratified cover construction used to produce the finite flat projective scheme cover Y over X with SNC boundary.","marker":"[KV04]"},{"why":"Introduces the irregular Hodge filtration on twisted de Rham cohomology whose degeneration is the paper's subject.","marker":"[Yu14]"},{"why":"Provides the quotient-stack structure theorem used to apply the Kresch–Vistoli cover to arbitrary smooth proper DM stacks.","marker":"[Kre09]"}],"fun_headline_variants":["E1 degeneration on stacks for irregular Hodge filtrations","Irregular Hodge filtrations degenerate on DM stacks","Stacks: E1 degeneration for irregular Hodge filtrations","E1 degeneration for irregular Hodge filtrations, any compactification","Irregular Hodge filtrations: E1 degeneration for all compactifications"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["E1 degeneration on stacks for irregular Hodge filtrations","Irregular Hodge filtrations degenerate on DM stacks","Stacks: E1 degeneration for irregular Hodge filtrations","E1 degeneration for irregular Hodge filtrations, any compactification","Irregular Hodge filtrations: E1 degeneration for all compactifications"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001601,"raw_usage":{"total_tokens":6331,"prompt_tokens":851,"completion_tokens":5480,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":5394}},"tokens_in":467,"tokens_out":5480,"duration_ms":37056,"temperature":1.0,"reasoning_tokens":5394,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:36:03.754505+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}