{"id":"c31416d3-47fd-4806-89d2-27502a106992","arxiv_id":"2608.07849","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Strict higher Du Bois singularities deform, satisfy base change for the relative Du Bois complex, and imply local freeness and Hodge-number constancy in families.","lead":"This paper proves that a class of singularities called strict higher Du Bois singularities stay stable under small deformations and satisfy base change for the relative Du Bois complex. It answers an open question of Kovács and Taji, and gives local freeness and constancy of Hodge numbers for families of singular varieties.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.2 is the sole engine for the deformation and base-change theorems, but its proof is a brief adaptation of an unpublished preprint and the key surjectivity in Proposition 5.14 is asserted without proof.","rationale":"The reader's weakest-assumption pinpoints Theorem 5.2 as the load-bearing point, and my reading agrees. I went through the rest of the paper looking for a more specific internal flaw. The examples in Section 9 are consistent: the apparent mismatch around H^1(S,Ω^1_S(2)) disappears once one tracks that A=O_S(2) makes A^1 correspond to the twist O_S(2), which is the allowed exception in Lemma 9.12(2). The Elkik-type lemmas and the right relative Du Bois construction are standard and appear sound. The only place where the paper is genuinely compressed is Section 5: the objects fffX_p(log KH) are not defined, Proposition 5.14's surjectivity is asserted, and Theorem 5.2 is delegated. Because the central theorems all depend on this, the manuscript should be accepted only conditional on either publishing [Kov26] or expanding the proof of Theorem 5.2 to be self-contained. My verdict remains CONDITIONAL, so no change from the reader.","tokens_in":36241,"tokens_out":29862,"duration_ms":286733,"concrete_test":"Write out the proof of Theorem 5.2 completely, following the structure of [Kov26, §8] with Ω^p_X in place of h^0(Ω^p_X). In particular, derive Proposition 5.14 from Lemma 5.13 by explicit computation of the map H^k(Y, fffY_p) → H^k(Y, fffY_p) and check surjectivity for each eigensheaf. Identify every step in the original argument that uses S2, torsion-freeness, or local duality on the sheaves being dualized, and either prove that strict-(m-1)-Du Bois implies the needed property for Ω^p_X or adjust the argument. If no complete proof can be supplied, the deformation and base-change theorems should be regarded as unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claims (Theorems 6.5, 7.2, 7.3, and consequently 8.7–8.8) all rest on Theorem 5.2, the higher Kovacs–Schwede injectivity for Kähler differentials. The proof of Theorem 5.2 is not self-contained: it says it follows 'essentially verbatim from [Kov26, §8]' with each h^0(Ω^p_X) replaced by Ω^p_X, and [Kov26] is an unpublished 2026 preprint. The adaptation is nontrivial because strict-(m-1)-Du Bois singularities (Definition 2.1(4)) do not include the S2/reflexivity conditions that are part of weak-m-Du Bois; the paper never proves that Ω^p_X has the depth or torsion-freeness that the original argument may use. The new input, Proposition 5.14, is proved in one sentence: it asserts a surjective map H^k(Y, fffY_p) → H^k(Y, fffY_p) on a cyclic cover Y and then says the functoriality of Ω^p_Y → Ω^p_Y gives the desired surjections. No justification is given for this surjectivity; it is not a formal consequence of functoriality. Since Proposition 5.14 is the Kähler replacement for [Kov26, Corollary 7.6], a gap here directly undermines Theorem 5.2. Without Theorem 5.2, the left inverse in Proposition 6.4 and hence the deformation theorem 6.5 cannot be concluded, and the affirmative answer to the Kovacs–Taji question (Theorem 7.3) is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new 'right' relative Du Bois complex and uses it to prove that strict-m-Du Bois singularities are invariant under small deformations, that strict-m-Du Bois fibers satisfy base change for the relative Du Bois complex, and that this yields local freeness of higher direct images and constancy of Hodge numbers. It also constructs examples showing that weaker m-Du Bois notions do not behave this way. The main theorems are Theorems 6.5, 7.2, 7.3, 8.7, and 8.8, with the Kovacs-Taji base-change question answered in the strict case. The overall architecture is attractive, but the proof currently has two load-bearing gaps: the injectivity theorem is imported from an unpublished preprint via a one-sentence adaptation, and the Hom-vanishing lemma at the heart of the base-change criterion contains a false cohomological amplitude claim.","tokens_in":36563,"tokens_out":12815,"duration_ms":133580,"significance":"If the proofs are completed, the results are substantial. The paper would establish deformation invariance for the strict higher Du Bois class, answer a base-change question of Kovacs-Taji, and extend the Friedman-Laza local-freeness theorem beyond the lci case. The right relative Du Bois complex is a useful new tool, and the sharpness examples in Section 9 are concrete and informative. The numerical constancy over arbitrary bases is a strong application. The main risk is not the plausibility of the statements but the state of the proofs: Theorem 5.2 depends on an unpublished preprint, Proposition 5.14 is asserted rather than proved, and Lemma 4.9 contains a specific incorrect vanishing argument. These are central to the deformation and base-change theorems rather than peripheral.","major_comments":[{"comment":"The proof of Theorem 5.2 is not self-contained and does not currently establish the theorem. The proof says it follows 'essentially verbatim from [Kov26, §8]' with h^0(Ω^p_X) replaced by Ω^p_X, but [Kov26] is an unpublished 2026 preprint and the replacement is not formal: strict-(m-1)-Du Bois (Definition 2.1(4)) is a different package from weak-m-Du Bois, which includes S2/reflexivity assumptions, and the manuscript gives no argument that Ω^p_X has the depth or torsion-freeness properties used in loc. cit. The new input Proposition 5.14 is the substitute for [Kov26, Corollary 7.6], yet its proof consists of one sentence asserting a surjective map H^k(Y, fff^Y_p) → H^k(Y, fff^Y_p) and claiming that functoriality of Ω^p_Y → Ω^p_Y decomposes it componentwise. No identification of this map or proof of its surjectivity is given, and functoriality alone does not imply surjectivity on hypercohomology. Since Theorem 6.5 and hence Theorems 7.2 and 7.3 rest on Theorem 5.2, this is a load-bearing gap.","section":"Section 5, Theorem 5.2 and Proposition 5.14"},{"comment":"The vanishing argument in Lemma 4.9 is incorrect as written. The proof claims that Ω^{p-2,+}_{X/B}[1] ⊗^L O_Z is 'supported in degree ≤ -1' and therefore has no maps to Ω^p_X|_Z. But Lemma 3.8 gives h^i(Ω^{p-2,+}_{X/B}) = 0 only outside [-p+2, m^+_{p-2}], with m^+_{p-2} ≥ 0; after the shift [1], cohomology can occur from degree -p+3 through m^+_{p-2}+1. For p = 3, Ω^{1,+}_{X/B}[1] has degree 0 term h^{-1}(Ω^{1,+}_{X/B}), which need not vanish (for smooth X it is isomorphic to O_X ⊗ ω_B), so the claimed Hom-vanishing Hom(Ω^{p-2,+}[1] ⊗ O_Z, Ω^p_X|_Z) = 0 is not justified. Since Theorem 4.11 and the 'if' direction of Corollary 4.12 use this vanishing, and Theorem 7.2 uses Corollary 4.12, this directly affects the main base-change theorem.","section":"Section 4, Lemma 4.9"},{"comment":"The deformation theorem depends on Proposition 6.4, whose final step is too compressed: after showing that Ω^p_Z → K_p → Ω^p_Z is the natural quasi-isomorphism, the proof asserts that 'the two outer ones extend to a left inverse of the middle map' without giving the triangulated-category argument or specifying the compatibility of the inverses on Ω^{p-1}_Z(-Z). This may be standard, but because the proposition is load-bearing for Theorem 6.5, the step should be written out in full.","section":"Section 6, Proposition 6.4 and Theorem 6.5"}],"minor_comments":[{"comment":"The typesetting makes Ω^p_X and Ω^p_X almost indistinguishable; please use a clearly visible underline or a different font for the Du Bois complex.","section":"Section 2, Definition 2.1"},{"comment":"The same symbol fff^Y_p is used for both the Kähler cofiltration and the Du Bois cofiltration in the statement and proof; distinct notation is needed to make the claimed surjection meaningful.","section":"Section 5, Proposition 5.14"},{"comment":"In condition (2), 'compatible with the map Ω^p_X → Ω^{p,*}_{X/B}' should specify which of the two maps from Ω^p_X is meant; the current wording can be read as requiring compatibility with two different maps.","section":"Section 3, Lemma 3.6"},{"comment":"The transition from the affine cone construction to a fiber of a morphism to A^1 is stated in one sentence at the end of Corollary 9.16; since the total space is affine, this is plausible, but the shrink should be made explicit.","section":"Section 9, Proposition 9.14"},{"comment":"There are several typographical and consistency issues: 'Ko´ acs' appears in the outline, the date says August 4 while the arXiv header says August 8, and the dependence on the unpublished preprints [Kov26] and [CDO26] should be flagged prominently in the introduction.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper has a promising architecture and the statements are plausible, but the proof currently has two load-bearing gaps: the injectivity theorem is inherited from an unpublished preprint in a non-formal way, and Lemma 4.9 contains a concrete erroneous cohomological amplitude claim. I recommend major revision rather than rejection because both issues are localized and could be repaired by expanding Section 5 and rewriting the Hom-vanishing step in Lemma 4.9. However, if Lemma 4.9 cannot be repaired, the proof of Theorem 7.2 collapses, so the authors should treat this as a priority."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper has a real new object (the right relative Du Bois complex), a clean base-change theorem for strict-m-Du Bois fibers, and a nice sharpness section. The main theorem answers Kovacs-Taji's question. I think it deserves a referee. But the proof of the engine, Theorem 5.2, is not self-contained, and one key surjectivity claim is asserted rather than proved.\n\nWhat I like: Construction 3.1 is a natural variant that builds the relative complex upward from p=0 instead of downward, and it pays off in Theorem 4.11 and the base-change criterion (Corollary 4.12). The deformation theorem (6.5) and the base-change theorem (7.2/7.3) are exactly the kind of result people have been after. Section 9's examples, especially the cone over S x P^1 with S a cubic surface (Prop. 9.14 and Cor. 9.16), show that weakening strict-m-Du Bois to m-Du Bois breaks both deformation and base change. That is a genuine sharpness result, not just an example.\n\nWhere I worry: the whole paper rests on Theorem 5.2, the higher Kovacs-Schwede injectivity for Kähler differentials. Its proof occupies one paragraph: 'follows essentially verbatim from [Kov26, §8]' with h^0 objects replaced by Ω^p_X. That could be fine if [Kov26] were published, but it's a 2026 preprint. More importantly, the adaptation is not formal: strict-(m-1)-Du Bois does not include the S2/reflexivity conditions used in weak-m-Du Bois, so depth/torsion-freeness of Ω^p_X has to be checked. The paper does not do that check.\n\nThe specific spot that bothers me is Proposition 5.14. It needs a surjection on hypercohomology of cofiltrations on a cyclic cover, and the proof says 'functoriality and equivariance' decompose it into the desired surjections. That's not a proof; surjectivity of an equivariant map does not automatically descend to eigensheaves without checking. The paper points to [Kov26, Corollary 6.6] for the Du Bois side, but the Kähler side is exactly the new ingredient. If this step is false, Theorems 6.5, 7.2, 7.3, and 8.7 all collapse.\n\nAlso, Lemma 4.9's Hom-vanishing argument is compressed, and Proposition 4.2 packs a lot into a few lines. These are probably fillable, but they are load-bearing.\n\nOverall: the architecture is coherent, the examples are explicit, and the paper is honest about the concurrency with [CDO26]. I don't see an internal contradiction, but I cannot certify the key injectivity theorem from what is written here. I'd send it to a serious referee with a request to expand Section 5 and to pin down the status of [Kov26] and [CDO26]. If that comes back clean, this is a significant paper.","headline":"Real contribution that answers Kovacs-Taji, but the proof of the central injectivity theorem is delegated to an unpublished preprint and one key surjection is asserted; referee should check Section 5 closely.","tokens_in":37124,"tokens_out":4278,"would_cite":true,"duration_ms":41171,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14B05","14D07","14F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Strict higher Du Bois singularities survive small deformations, and base change follows.","keywords":["strict-m-Du Bois singularities","higher Du Bois singularities","relative Du Bois complex","base change","deformation invariance","local freeness","Hodge numbers","Kähler differentials"],"falsifier":"For a concrete strict-$1$-Du Bois fiber $X_b$ of a flat morphism to a smooth curve, compute the derived restriction $\\Omega^{1,+}_{X/B} \\otimes^{\\mathbf{L}} \\mathcal{O}_{X_b}$; if it is not quasi-isomorphic to $\\Omega^1_{X_b}$, the main theorem fails. Alternatively, test the surjectivity $H^k(X, \\mathrm{fff}^p_X(\\log K_H)\\otimes L^{-i}) \\to H^k(X, \\mathrm{fff}^p_X(\\log H)\\otimes L^{-i})$ on a cyclic cover of a non-lci strict-$1$-Du Bois variety; a single degree where surjectivity fails would break Theorem 5.2 and with it Theorem 6.5.","tokens_in":35989,"feed_emoji":"📐","tokens_out":10517,"duration_ms":95738,"temperature":0.7,"pith_summary":"This paper establishes that strict higher Du Bois singularities, the most direct generalization of Du Bois singularities to Kähler $p$-forms for $p\\le m$, are stable under small deformations: if a Cartier divisor in a complex variety is strict-$m$-Du Bois, then the ambient variety is strict-$m$-Du Bois near that divisor. Using this stability, it proves a base-change theorem: for a fiber of a morphism to a smooth curve, strict-$m$-Du Bois singularities force the relative Du Bois complex to restrict to the absolute Du Bois complex of the fiber in degrees up to $m$, and strict fibers of every order give full base change for both the left and right relative Du Bois complexes. This answers a question left open in earlier work and has concrete consequences for families: the higher direct images of relative Kähler differentials are locally free over a smooth curve, and Hodge numbers are constant in flat families over arbitrary bases. The paper also constructs counterexamples showing that the weaker $m$-Du Bois condition does not suffice, so the strictness condition is essentially sharp.","feed_headline":"Strict higher Du Bois singularities survive small deformations","feed_subtitle":"Base change now follows, bringing local freeness, constant Hodge numbers, and sharp counterexamples without strictness.","key_machinery":"The engine is the right relative Du Bois complex $\\Omega^{p,+}_{X/B}$, built upward from $p=0$ by a cone construction using the wedge maps $\\Omega^p_X \\otimes f^*\\omega_B \\to \\Omega^{p+1}_X$ (Construction 3.1 and Theorem 3.2). Its main advantage is a commuting diagram (Theorem 4.11) that compares the derived restriction $\\Omega^{p,+}_{X/B} \\otimes^{\\mathbf{L}} \\mathcal{O}_{X_b}$ with $\\Omega^p_X \\otimes^{\\mathbf{L}} \\mathcal{O}_{X_b}$ and $\\Omega^p_{X_b}$, reducing base change to exactness of the conormal triangle. The deformation theorem is carried by a higher injectivity theorem for Kähler differentials (Theorem 5.2), built from cyclic covers and divisorial log structures; that injectivity, combined with a lifting lemma for distinguished triangles, converts left inverses on the fiber into quasi-isomorphisms on the ambient variety. The cohomological amplitude properties of the right complex are what make the key diagram commute.","core_discovery":"On the paper's own terms, the central discovery is that strict-$m$-Du Bois is the correct level of higher Du Bois behavior for deformation and base-change questions. Theorem 6.5 says that a Cartier divisor $Z$ with strict-$m$-Du Bois singularities forces $X$ to have strict-$m$-Du Bois singularities in a neighborhood of $Z$. From this, Theorems 7.2 and 7.3 derive right-$m$-base change for the right relative Du Bois complex and full base change for both relative complexes when the fiber is strict for every $m$, i.e. $\\Omega^{p,*}_{X/B} \\otimes^{\\mathbf{L}} \\mathcal{O}_{X_b} \\simeq \\Omega^p_{X_b}$ for all $p$. The applications are local freeness of $R^i f_* \\Omega^p_{X/B}$ for flat proper families over a smooth curve and constancy of Hodge numbers over arbitrary bases, while Section 9 gives families where $1$-Du Bois fibers break both base change and deformation invariance.","pith_inferences":["The same cone-based mechanism should yield base-change statements for relative logarithmic Du Bois complexes under the same strictness assumptions.","One can test whether the deformation theorem extends to non-Cartier divisors: if $Z$ is only $S_2$ and of codimension two, inversion-of-adjunction style arguments might force strict-$m$-Du Bois on $X$ away from a small set.","The sharpness examples suggest that any base-change-compatible higher Du Bois theory must treat the injectivity property as part of the definition rather than as a consequence."],"forward_implications":["Strict-$m$-Du Bois singularities are deformation invariant: any family with a strict-$m$-Du Bois special fiber has strict-$m$-Du Bois nearby fibers.","A strict-$m$-Du Bois fiber satisfies right-$m$-base change, so the right relative Du Bois complex computes the fiber's Du Bois complex in degrees up to $m$.","A fiber that is strict-$m$-Du Bois for every $m$ satisfies full base change for both left and right relative Du Bois complexes.","For flat proper families over a smooth curve, higher direct images of relative Kähler differentials are locally free and compatible with base change up to degree $m$ when a fiber is strict-$m$-Du Bois.","Hodge numbers of fibers are constant in flat families over arbitrary bases under the same strictness hypothesis, while without strictness, base change and deformation invariance fail already for $1$-Du Bois fibers."],"supporting_citations":[{"why":"Supplies the higher injectivity theorem whose proof is adapted to Kähler differentials in Theorem 5.2.","marker":"[Kov26]"},{"why":"Constructs the left relative Du Bois complex and frames the base-change question answered in Theorem 7.3.","marker":"[KT25]"},{"why":"Gives the original relative Du Bois complex construction that the right complex reverses.","marker":"[Kov96]"},{"why":"Provides the definitions of higher Du Bois conditions and the Bertini-type lemma used to spread strictness to nearby fibers.","marker":"[SVV23]"},{"why":"Proves base change for general members and supplies the earlier failure example that the sharpness results extend.","marker":"[JK25]"},{"why":"Supplies the local-freeness criterion whose flatness hypothesis is verified here by base change.","marker":"[FL24]"},{"why":"Computations of Du Bois complexes on cones are used in the 1-Du Bois counterexample.","marker":"[PS25]"},{"why":"Provides the residue sequences and cyclic-cover decompositions underlying the injectivity theorem.","marker":"[EV92]"},{"why":"Foundational construction of the Du Bois complex whose graded pieces are the objects studied here.","marker":"[DB81]"},{"why":"Contributes the deformation lifting lemmas and injectivity framework used in Theorem 6.5.","marker":"[KS16a]"}],"fun_headline_variants":["Strict higher Du Bois: invariant and base change","Base change for strict Du Bois, local freeness applies","Du Bois base change: sharp condition is strictness","Small deformations keep Du Bois strict, base change works","Higher Du Bois: deformation invariance, base change, Hodge"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the higher injectivity theorem for Kähler differentials (Theorem 5.2), whose proof is an adaptation of an argument from an unpublished preprint; if that argument or the new surjectivity input (Proposition 5.14) fails, the deformation theorem and the base-change consequences collapse.","fun_headline_variants_meta":{"raw":{"variants":["Strict higher Du Bois: invariant and base change","Base change for strict Du Bois, local freeness applies","Du Bois base change: sharp condition is strictness","Small deformations keep Du Bois strict, base change works","Higher Du Bois: deformation invariance, base change, Hodge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1598,"prompt_tokens":863,"completion_tokens":735,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":665}},"tokens_in":479,"tokens_out":735,"duration_ms":7470,"temperature":1.0,"reasoning_tokens":665,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:46:20.034769+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete strict-$1$-Du Bois fiber $X_b$ of a flat morphism to a smooth curve, compute the derived restriction $\\Omega^{1,+}_{X/B} \\otimes^{\\mathbf{L}} \\mathcal{O}_{X_b}$; if it is not quasi-isomorphic to $\\Omega^1_{X_b}$, the main theorem fails. Alternatively, test the surjectivity $H^k(X, \\mathrm{fff}^p_X(\\log K_H)\\otimes L^{-i}) \\to H^k(X, \\mathrm{fff}^p_X(\\log H)\\otimes L^{-i})$ on a cyclic cover of a non-lci strict-$1$-Du Bois variety; a single degree where surjectivity fails would break Theorem 5.2 and with it Theorem 6.5.","supporting_citations":[],"review_version":1}