{"id":"d9e3e582-76ea-4280-8ea4-1ff54f10a586","arxiv_id":"2501.14065","paper_version":2,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The Hodge rational homology level HRH(Z) generalizes Q-homology manifolds and is characterized by local cohomology, link cohomology, and V-filtration conditions.","lead":"This paper introduces an invariant HRH(Z) that measures how far a singular complex variety is from behaving like a rational homology manifold in each Hodge degree. It then shows that HRH(Z) is detected by local cohomology, link invariants, and V-filtration data, and uses it to obtain partial Poincaré duality and new bounds on singular loci.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The advertised interpretation of HRH as encoding the Du Bois/rational gap rests on Saito's unpublished quasi-isomorphism from [Sai99], which is used without proof; if it fails for some reduced varieties, the invariant in the main theorems is not the Du Bois-theoretic one advertised.","rationale":"The reader's weakest assumption identifies Saito's quasi-isomorphism from [Sai99] as load-bearing because it converts the Hodge-theoretic duality morphism into maps between Du Bois complexes, which is precisely the step connecting HRH to singularities and local cohomology. I agree with this assessment. My reading of the paper confirms that the main theorems (B, D, E, G) are proven for the Hodge-theoretic invariant defined in Definition 4.3 via ψ_Z, and they do not explicitly rely on the Du Bois identification; however, the paper's advertised purpose and the comments connecting HRH to higher Du Bois and higher rational singularities (e.g., Remark 4.8, Definition 3.2) depend on the identification. I did not find an internal contradiction in the proofs, but the dependence on [Sai99] is not merely cosmetic: it is the bridge between the invariant being studied and the singularities it is claimed to measure. A concrete test on normal crossing divisors or cones is feasible because both sides are computable in those cases. The existing CONDITIONAL verdict already reflects the need for specialist confirmation, so my concern does not move the verdict; it sharpens the reason for the condition.","tokens_in":55698,"tokens_out":15366,"duration_ms":140731,"concrete_test":"Compute both sides of the quasi-isomorphism Ω^p_Z[d-p] ≅ Gr^F_{-p}DR_Z(Q^H_Z[d]) for a reduced but non-normal variety where both are explicitly computable, for example a simple normal crossing divisor (union of two smooth components meeting transversely) or an affine cone over a smooth projective variety. For a normal crossing divisor, Ω^p_Z is the p-th exterior power of the logarithmic forms with the appropriate torsion-free quotient, while Gr^F_{-p}DR_Z(Q^H_Z[d]) can be computed from the standard mixed Hodge module structure on a normal crossing variety. If the two agree for p = 0, 1, 2 in these examples, the identification is supported; if a mismatch appears, the advertised Du Bois interpretation of HRH is not valid in that class.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper defines HRH in the introduction via morphisms φ_p between Du Bois complexes, using the identification Ω^p_Z[d-p] ≅ Gr^F_{-p}DR_Z(Q^H_Z[d]) from the unpublished preprint [Sai99]. This identification is what turns the Hodge-theoretic duality morphism ψ_Z into statements about Du Bois complexes, and it is exactly what connects HRH to higher Du Bois and higher rational singularities. The main theorems (B, D, E, G) are proven for the Hodge-theoretic Definition 4.3 directly in terms of ψ_Z, so they do not explicitly invoke [Sai99]; however, the abstract's claim that HRH 'encodes the difference between higher Du Bois and higher rational singularities for local complete intersections' and the comparison with condition (*)_k of [PP24] in Remark 5.2 depend on this identification. Since [Sai99] is a preprint and the paper cites it without proof or qualification, the geometric meaning of HRH is at risk if the quasi-isomorphism fails for some reduced pure-dimensional varieties, especially non-normal or non-Cohen-Macaulay ones. The concern is not an internal inconsistency but an unverified external foundation for the central interpretation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an invariant HRH(Z), the Hodge rational homology level, defined by requiring that the graded de Rham pieces of Saito's Poincaré duality morphism ψ_Z be quasi-isomorphisms up to Hodge degree k. In the introduction this is equivalently phrased through the morphisms φ_p between Du Bois complexes, and the authors claim the invariant encodes the difference between higher Du Bois and higher rational singularities for local complete intersections. The main results are: an embedded characterization via filtered local cohomology mixed Hodge modules (Theorem B), pointwise characterizations via local cohomology and link invariants (Theorems D, E, F), a partial Poincaré duality statement (Theorem A), a bound involving a new generic local cohomological defect lcdef_gen(Z) (Theorem G), and a series of results in the local complete intersection case relating HRH to the V-filtration, the unipotent specialization quotient Q_Z, spectral numbers, and Bernstein-Sato polynomials (Theorems H--K, Corollary L). The paper closes with extensive examples: affine cones, toric varieties, secant varieties, determinantal varieties, Thom-Sebastiani examples, and liminal singularities.","tokens_in":55926,"tokens_out":15885,"duration_ms":137866,"significance":"If the main characterizations are correct, HRH(Z) is a local, computable invariant that gives a Hodge-theoretic refinement of rational homology manifolds and interfaces with the recently studied higher Du Bois and higher rational singularity classes. The paper contains substantial explicit computations, particularly for determinantal varieties, and proposes a concrete improvement of the codimension bound of Park--Popa via lcdef_gen(Z). The conjectural statement about Bernstein-Sato polynomials (Conjecture 11.20) and the worked examples are useful and falsifiable. The significance is conditional, however, on two external foundations: the identification of the de Rham graded pieces of the trivial Hodge module with Du Bois complexes is cited to the unpublished preprint [Sai99], and several structural results are imported from the arXiv preprint [PP24] via the equivalence in Remark 5.2.","major_comments":[{"comment":"The identification Ω^p_Z[d-p] ≅ Gr^F_{-p}DR_Z(Q^H_Z[d]) is invoked from the unpublished preprint [Sai99] and is load-bearing for the advertised interpretation of HRH as measuring the Du Bois/rational gap. If this quasi-isomorphism fails for some reduced pure-dimensional varieties (for instance non-normal or non-Cohen-Macaulay ones), then Definition 0.1 and the statements of Theorems B, D, E, and G do not concern the Du Bois-theoretic invariant the abstract describes. Please include a proof or a precise published reference for this identification, or add an explicit hypothesis under which it is known; otherwise the geometric meaning of HRH is not established.","section":"Section 3, Remark 4.4, Definition 0.1"},{"comment":"The definition of HRH in Definition 4.3 checks Gr^F_{-p}DR_Z(ψ_Z) for p ≤ k, i.e. high Hodge filtration pieces. In contrast, §5 and the proof of Theorem B check Gr^F_{p-d}DR_X(i_*ψ_Z) for p ≤ k and then invoke Corollary 1.7 to conclude that HRH(Z) ≥ k is equivalent to F_{k-d}i_*ψ_Z being a quasi-isomorphism. These two conditions are not equivalent as written: Corollary 1.7 with F_{k-d} would require the pieces Gr^F_ℓ for ℓ ≤ k-d, which are low filtration pieces, while Definition 4.3 uses the pieces with index -p for p ≤ k. The target indices d-p and p-d are also opposite. This discrepancy affects Theorem B and Theorem H, both of which use the §5 convention. Please reconcile the definition with the proof, or state explicitly that Definition 4.3 uses the opposite Hodge-filtration convention.","section":"Definition 4.3 vs. §5 and proof of Theorem B"},{"comment":"The proof of Proposition 8.5 uses the inequality HRH_x(Z) ≤ HRH_x(Z∩T_α) for a normal slice T_α without proof or citation. This inequality is not a formal consequence of Lemma 6.10, since the normal slice embedding is not a smooth morphism. If it is intended to follow from repeated general hyperplane sections via Remark 4.8(2), that argument should be supplied, because Proposition 8.5 is the basis for Theorem G.","section":"Proof of Proposition 8.5"}],"minor_comments":[{"comment":"The introduction says the generic local cohomological defect lcdef_gen(Z) is introduced in §5, but Definition 8.1 is in §8; the outline also assigns Theorem G (= Proposition 8.5) to §5 and then again to §8. Please correct these cross-references.","section":"Introduction and outline"},{"comment":"The proof of Corollary 5.4 says the assertion is immediate from [Ola23, (6.1)], but the Hodge-filtration shifts connecting F_{p-n}Gr^W_{n+l}H^1_D(O_X) to the weighted Hodge ideals are not shown. A short explicit derivation would improve readability.","section":"Corollary 5.4"},{"comment":"The Macaulay2 computation in Example 11.21 is cited as evidence for the Bernstein-Sato polynomial identity, but the input and output are not included. Please provide the exact code or a reproducible verification, since this example is used to illustrate the dichotomy in Conjecture 11.20.","section":"Example 11.21"},{"comment":"Several load-bearing references are arXiv preprints, including [Sai99], [PP24], [Dir23], and [CDM22]. For the final version, please indicate the publication status of each and, where possible, cite the published version.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's relation to [PP24] and [PSV24] is central: Remark 5.2 states an equivalence with condition (*)_k, and Lemma 6.2 and Remark 4.8 import results from [PP24]. The editor may wish to check that the novelty boundary with those preprints is clearly delineated. The main unresolved technical risk is the unpublished identification [Sai99]; even if the main theorems are formally about the Hodge-theoretic ψ_Z, the paper's advertised interpretation depends on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core invariant HRH(Z) is not new — it is condition (*)_k from Park-Popa, and the authors say so in Remark 5.2. What is new are the characterizations: Theorem B gives an embedded local cohomology criterion, Theorem D gives a pointwise criterion via H^*_{x}(Z), Theorem E/F connect to link invariants, and Theorem G introduces lcdef_gen and improves the PP24 codimension bound. The LCI section (Theorems H–K) gives several integer invariants and inequalities, and the examples — cones, determinantal varieties, Thom–Sebastiani, liminal — are genuinely useful. The paper is well-written, the proofs are detailed, and the comparison with PP24 is honest.\n\nThe main soft spot is the identification of the Du Bois complex with the associated graded de Rham functor applied to Q^H_Z[d], attributed to Saito's unpublished preprint [Sai99]. The abstract and intro sell HRH as encoding the Du Bois/rational gap, and that interpretation depends on this identification. The formal Definition 4.3 and the proofs of Theorems B, D, E, G work directly with the Hodge-theoretic morphism ψ_Z, so they do not rely on [Sai99]. But the advertised geometric meaning does. If [Sai99] fails for some reduced non-normal or non-CM varieties, the invariant in the main theorems is still well-defined, but it is not the Du Bois-theoretic one advertised. This is worth flagging to a referee, but it is a known result in the field, so I would not treat it as fatal. A second concern is that several technical inputs are from arXiv preprints by the same authors ([Dir23], [CDM22], [Ola23]). That is common in this area, but it does mean some parts need same-group verification.\n\nThe stress-test note is on point but I would temper it: the main theorems are about ψ_Z, and the [Sai99] issue only affects the interpretation, not the internal logic. The reader's CONDITIONAL verdict and low confidence are fair, though I would give slightly more credit to the examples and the lcdef_gen result.\n\nWho is this for? People working on higher Du Bois/rational singularities, local cohomology, and mixed Hodge modules. They will get real value from the characterizations and the computations. It deserves a serious referee, ideally a specialist in Saito's theory who can verify the use of [Sai99] and the same-author preprints. I would accept it for peer review.","headline":"A solid, useful paper that gives new characterizations and examples for an invariant equivalent to Park-Popa's condition (*)_k; the main caveat is that the advertised Du Bois interpretation depends on an unpublished Saito result.","tokens_in":56521,"tokens_out":2890,"would_cite":true,"duration_ms":25290,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14B05","14F10","32S35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new invariant HRH(Z) measures how far a singular variety is from a rational homology manifold, and the paper proves it is detected by filtered local cohomology at each point.","keywords":["Hodge rational homology level","rational homology manifolds","mixed Hodge modules","Du Bois singularities","local cohomology","Poincaré duality","local complete intersections","Bernstein-Sato polynomial"],"falsifier":"Take an affine cone Z over a smooth projective variety X with an ample line bundle L, choose k where the paper's Proposition 13.1 says HRH(Z)<k because the cup product $H^{{k-1}}$($Ω^{{k-1}}$_X) → H^k(Ω^k_X) fails to be an isomorphism, and compute F_{k-d}H^i_{\\{v\\}}(Z) at the cone vertex. The claim is that these groups vanish except for i=2d, where the piece is Q; observing otherwise would falsify Theorem 6.5.","tokens_in":55465,"feed_emoji":"📐","tokens_out":8995,"duration_ms":74187,"temperature":0.7,"pith_summary":"This paper defines HRH(Z), the Hodge rational homology level of a pure-dimensional complex variety Z, which measures how many Hodge-filtered pieces of the Poincaré duality morphism are isomorphisms. HRH(Z)=∞ recovers the classical notion of a rational homology manifold, while finite values quantify the gap between higher Du Bois and higher rational singularities. The central result is that HRH(Z)≥k can be checked locally: in the embedded setting it is equivalent to vanishing and equality conditions on the filtered local cohomology modules of the ambient structure sheaf, and pointwise it is equivalent to the filtered local cohomology of Z at each point, hence to data on the cohomology of the link. From this the authors derive partial Poincaré duality isomorphisms, a bound on the locus where Z fails to be a rational homology manifold, and, for local complete intersections, comparisons with the minimal exponent, the integral spectrum, and Bernstein–Sato roots.","feed_headline":"Filtered local cohomology computes a variety's Hodge-homology level","feed_subtitle":"The new invariant HRH(Z) generalizes rational homology manifolds and separates higher Du Bois from higher rational singularities.","key_machinery":"The central object is the Hodge-theoretic Poincaré duality morphism ψ_Z: Q^H_Z[d] → D_Z(Q^H_Z[d])(−d) in the derived category of mixed Hodge modules on Z, together with its associated graded de Rham functor. Applying Gr^F_{−p}DR_Z turns ψ_Z into morphisms φ_p between Du Bois complexes and their Grothendieck duals, so the invariant HRH(Z) counts how many of these φ_p are quasi-isomorphisms. The identification Ω^p_Z[d−p] ≅ Gr^F_{−p}DR_Z(Q^H_Z[d]) is the bridge that lets local cohomology mixed Hodge modules, V-filtrations, and link cohomology compute HRH(Z).","core_discovery":"The paper's central claim is that the correct Hodge-theoretic generalization of a rational homology manifold is the condition that the morphisms φ_p: Ω^p_Z → D_Z($Ω^{{d-p}}$_Z)[-d] obtained from the Poincaré duality morphism ψ_Z are quasi-isomorphisms for all p≤k, and that this condition is computable from filtered local cohomology. Theorem B states that for a closed embedding of a pure c-codimensional variety Z in a smooth n-dimensional X, HRH(Z)≥k if and only if F_{k-n}H^j_Z(O_X)=0 for all j>c and F_{k-n}W_{n+c}H^c_Z(O_X)=F_{k-n}H^c_Z(O_X). Theorem 6.5 sharpens this to points: HRH(Z)≥k if and only if for every x∈Z, F_{k-d}H^i_{\\{x\\}}(Z)=0 for i<2d and F_{k-d}$H^{{2d}}$_{\\{x\\}}(Z)=Q. A rational homology manifold is exactly the case HRH(Z)=∞, and when HRH(Z)<∞ it is bounded above by (d−3)/2. For local complete intersections, the paper shows HRH(Z) is controlled by the V-filtration: for hypersurfaces HRH(Z)=p($Gr^{0}$_V(B_f))+n−2, and in general inequalities relate HRH(Z) to the integer invariants p(Q_Z,F), Sp_{min,Z}(Z,x), and the reduced Bernstein–Sato polynomial.","pith_inferences":["Editorial inference: because Theorem B expresses HRH in terms of filtered local cohomology modules, the invariant is in principle computable by D-module algorithms whenever the V-filtration on those modules can be computed, making HRH a practical singularity detector.","Editorial inference: the pointwise reformulation suggests a Hodge-filtered analogue of the known irreducibility criterion for rational homology manifolds, and one might expect HRH_x(Z)≥k to control low-degree Betti numbers of the link beyond the range stated in Theorem F.","Editorial inference: the inequality lcdef_gen(Z)+2HRH(Z)+3≤codim_Z(Z_{nRS}) could serve as a test for whether known classes of singularities, such as secant varieties or finite group quotients, achieve the sharp codimension bound; the paper computes equality in several determinantal examples.","Editorial inference: the relation to weighted Hodge ideals in Corollary 5.4 suggests that HRH of hypersurfaces can be probed by multiplier-ideal-type computations, potentially extending known minimal-exponent bounds to non-isolated hypersurface singularities."],"forward_implications":["If HRH(Z)=∞ recovers rational homology manifolds, then every rational homology manifold satisfies all the partial Poincaré duality isomorphisms, and every non-rational one has HRH(Z) ≤ (d−3)/2.","Varieties with HRH(Z)≥k admit filtered Poincaré duality isomorphisms F_{d−k}H^{d−i}(Z) ≅ F^{−k}H^{d+i}_c(Z)^∨ for all i.","HRH(Z)≥0 forces the variety to be irreducible at each point and bounds the codimension of the non-rational-homology-manifold locus from below by 2HRH(Z)+3, an inequality sharpened by the new generic local cohomological defect lcdef_gen(Z).","Higher rational singularities imply HRH(Z)≥k; for local complete intersections, HRH(Z)≥k separates k-Du Bois from k-rational behavior, and in the hypersurface case HRH is exactly computable from the V-filtration.","For local complete intersections, HRH(Z) is bounded below by integer invariants attached to the specialization module Q_Z and the integral spectrum, and at isolated hypersurface singularities HRH(Z)=Sp_{min,Z}(Z,x)−2."],"supporting_citations":[{"why":"Constructs the Hodge-theoretic Poincaré duality morphism ψ_Z and the mixed Hodge module formalism used throughout.","marker":"[Sai90]"},{"why":"Supplies the quasi-isomorphism Ω^p_Z[d−p] ≅ Gr^F_{−p}DR_Z(Q^H_Z[d]) that connects HRH to Du Bois complexes.","marker":"[Sai99]"},{"why":"Provides the filtered de Rham functor properties and strictness results used to pass from ψ_Z to φ_p.","marker":"[Sai88]"},{"why":"Establishes the Hodge filtration on local cohomology modules and its relation to Du Bois complexes that Theorem B and Corollary C rely on.","marker":"[MP22]"},{"why":"Defines the equivalent (*)_k condition and gives the codimension and Hodge symmetry constraints that the paper extends.","marker":"[PP24]"},{"why":"Introduces the link invariants ℓ_{p,q} and the higher rational and Du Bois framework used in Theorem E.","marker":"[FL24a]"},{"why":"Relates the V-filtration and pole-order filtration on local cohomology to k-rational singularities, the model for the embedded characterization.","marker":"[CDM22]"},{"why":"Supplies the microlocalization and spectrum lemmas for the module Q_Z used in the local complete intersection results.","marker":"[Dir23]"},{"why":"Defines the integral spectrum Sp(Z,x) and the product formula for Verdier specialization used in the LCI comparisons.","marker":"[DMS11]"},{"why":"Establishes the hypersurface higher Du Bois theory and the equality between minimal spectral number and HRH in isolated cases.","marker":"[JKSY22]"}],"fun_headline_variants":["Filtered local cohomology computes Hodge-homology level","HRH(Z) sharpens rational homology via Hodge theory","Local cohomology yields Hodge generalization of Q-manifolds","HRH(Z): Hodge homology level for singular varieties","Filtered local cohomology computes HRH(Z) for Q-manifolds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's load-bearing premise is that the p-th Du Bois complex of Z, a Hodge-theoretic replacement for Kähler differentials, is identified with the (−p)-th graded piece of the de Rham functor of the trivial Hodge module; if that identification fails for some reduced variety, the invariant HRH(Z) no longer tracks Du Bois singularities and the main characterizations collapse.","fun_headline_variants_meta":{"raw":{"variants":["Filtered local cohomology computes Hodge-homology level","HRH(Z) sharpens rational homology via Hodge theory","Local cohomology yields Hodge generalization of Q-manifolds","HRH(Z): Hodge homology level for singular varieties","Filtered local cohomology computes HRH(Z) for Q-manifolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00068,"raw_usage":{"total_tokens":3148,"prompt_tokens":1066,"completion_tokens":2082,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":682,"completion_tokens_details":{"reasoning_tokens":1991}},"tokens_in":682,"tokens_out":2082,"duration_ms":14650,"temperature":1.0,"reasoning_tokens":1991,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:24:56.572435+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an affine cone Z over a smooth projective variety X with an ample line bundle L, choose k where the paper's Proposition 13.1 says HRH(Z)<k because the cup product $H^{{k-1}}$($Ω^{{k-1}}$_X) → H^k(Ω^k_X) fails to be an isomorphism, and compute F_{k-d}H^i_{\\{v\\}}(Z) at the cone vertex. The claim is that these groups vanish except for i=2d, where the piece is Q; observing otherwise would falsify Theorem 6.5.","supporting_citations":[],"review_version":1}