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On higher Du Bois singularities and $K$-regularity

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Pith's one-line read Higher Du Bois singularities are shown to be equivalent, in many characteristic-zero settings, to K-regularity, yielding a strengthened Vorst conjecture for local complete intersections and a projective characterization of K-regularity.

desk verdict A serious paper with a strong affine core and a plausible but under-supported projective characterization; the gap in Theorem 7.1's proof should keep it from being approved as-is. read the letter →

arxiv 2504.12402 v3 pith:U5O6UPKE submitted 2025-04-16 math.AG math.KT

Wanchun Shen This is my paper · ORCID
classification math.AGmath.KT
keywords boisregularityhigherrelationshipsingularitiesalgebraicbuildingcharacteristic
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The pith

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The reading

Algebraic K-theory attaches abelian groups K_m(X) to a space X. For a smooth space, these groups do not change when X is multiplied by a line. Singular spaces can break this property, and measuring the failure is what K-regularity does. The paper connects this K-theoretic measurement to Du Bois complexes, Hodge-theoretic objects that detect how badly a space is singular. For local complete intersections, a mild class of singular spaces, the paper shows that Du Bois conditions at level m match K-regularity at a specific negative index. It proves a strengthened Vorst conjecture: an affine local complete intersection that is K_2-regular must actually be smooth. For hypersurfaces with isolated singularities, the criterion can be written with a number called the minimal exponent: K_m-regularity holds exactly when the minimal exponent is at least (m+d)/2 rounded up. The paper also gives a characterization for projective varieties: K_m-regularity is equivalent to certain cohomology maps between derived and ordinary differential forms being isomorphisms. Section 6 classifies the Du Bois table of surfaces, showing the two nonzero invariants can take any ordered pair of integers, and applies this to Bass's question about A^1 versus A^2 invariance. The bridge runs through the derived de Rham complex and its Hodge filtration. The proofs use vanishing theorems for Du Bois complexes to upgrade the equivalence in the affine case and to prove new regularity criteria. All main theorems are stated over fields of characteristic zero.
Extended reading notes

Core claim

The load-bearing assertion is Theorem 7.1: for a projective variety X over a field F of characteristic zero, X is K_m-regular if and only if the natural maps H^i(X,L^p_{X/F}) to H^i(X,Omega^p_{X/F}) are isomorphisms for all i-p >= -m+1. This theorem underpins the projective examples and, together with the CHSW and CHWW machine, yields the strengthened Vorst statement: an affine local complete intersection that is K_2-regular is regular (Theorem B and Theorem F).

Load-bearing premise

The proof of Theorem 7.1 reduces to F=C by 'standard base change arguments' and then invokes Hodge-to-de Rham degeneration to obtain the surjectivity in Proposition 7.3. The fragile premise is that the cohomological isomorphism conditions over C descend back to the original field F through the spectral sequences of Lemma 2.10(1); the text itself admits that the direct implication at the Q-level is unclear and proceeds by a roundabout spectral sequence argument. If this descent or the roundabout verification fails, the projective characterization in Theorem I loses its foundation.

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Referee Report

3 major / 4 minor

Summary. The paper develops a dictionary between higher Du Bois singularities and K_m-regularity for varieties over fields of characteristic zero. The main affine results are: Proposition C and Theorem D, which connect m-Du Bois/pre-m-Du Bois conditions to K-regularity in both directions; Corollary E, an equivalence for local complete intersections with isolated singularities; Theorem F, which upgrades Vorst's conjecture for affine local complete intersections by showing that K_2-regularity (or K_1-regularity, depending on the parity of the codimension of the singular locus) forces regularity; and Theorem A, a numerical criterion for hypersurfaces via the minimal exponent. The paper also contains a study of Bass's question using Du Bois invariants of surfaces, including a claim that all pairs (b_{0,1}, b_{1,1}) with b_{0,1} >= b_{1,1} occur, and a projective characterization of K_m-regularity (Theorem 7.1 = Theorem I) that is used in two examples.

Significance. If the results are correct, the paper makes a substantial contribution: it gives the first systematic bridge between higher Du Bois singularities and algebraic K-theory, proves a strong form of Vorst's conjecture for local complete intersections in characteristic zero, and produces new tools for constructing and detecting K_m-regular varieties. The affine Vorst strengthening is particularly valuable because it is proved using the established CHWW/CHSW machine plus vanishing theorems for Du Bois complexes, and it does not appear to depend on the projective Theorem 7.1. The paper also gives useful examples and a surface-level classification of Du Bois invariants. However, the projective characterization Theorem 7.1 is not established as written: its proof contains a missing base-change justification and an incomplete diagram argument, and the text itself flags the unclear step. Since Theorem 7.1 is stated as a main theorem and is used in Examples 4.3 and 4.5, the paper needs substantive repair before the projective statements can be accepted.

major comments (3)
  1. [§7 (Proposition 7.3 and Theorem 7.1)] The proof of Theorem 7.1 is incomplete as written. The reduction to F=C in Proposition 7.3 is justified only by 'standard base change arguments' referring to Lemma 2.10; Lemma 2.10 provides spectral sequences relating the L^p and Ω^p objects over different fields, but no statement in the paper establishes that the surjectivity or isomorphy of the displayed hypercohomology maps over C descends to the original field F. Second, the commutative diagram in the proof of Proposition 7.3 omits the vertical maps: from the surjectivity of H^i(X,C) -> H^i(X,Ω^{≤p}_{X/C}) one cannot conclude that H^i(X,LΩ^{≤p}_{X/C}) -> H^i(X,Ω^{≤p}_{X/C}) is surjective. The natural candidate for the missing arrow, H^i(X,C) -> H^i(X,LΩ^{≤p}_{X/C}), fails in general: for a cuspidal projective cubic and p=0, H^1(X,C)=0 while H^1(X,O_X)=H^1(X,LΩ^{≤0}_{X/C}) is one-dimensional. The text itself notes in footnote 4 that the direct implication is unclear and takes a 'roundabout' spectral-sequence route, but that route rests on Proposition 7.3, so the missing justification is load-bearing. Consequently Theorem I, Remark 7.2, and the projective examples (Examples 4.3 and 4.5) are not established as written.
  2. [§5 (proof of Theorem F, part (2))] The proof of part (2) of Theorem F begins 'Suppose codim X_sing = 2m+1 is odd', even though the statement of part (2) assumes that codim X_sing is even. The displayed inequalities that follow (for example m ≤ codim X_sing - 2 = 2m - 2) are the even-case computation. As printed, the proof does not cover the stated even-codimension case; it should begin with codim X_sing = 2m and then recheck the inequalities, especially the boundary case m=1.
  3. [§6 (Proposition 6.3(2))] Proposition 6.3(2) is under-specified. The hypersurface is written as x^3+y^3+z^{3m}+xyz^{m+a} (or possibly with a separate additive constant a), but the parameter a is not defined, and the statement claims b_{1,1}=n while the proof concludes b_{1,1}=a. The proof also treats a=0 and a≥1 separately and invokes a deformation argument for the independence of b_{0,1} from a. Please define the family f_a precisely, state the relation between a and n (presumably a=n), and spell out the deformation argument before claiming the classification for all pairs m≥n≥0.
minor comments (4)
  1. [§4 (Corollary 4.8)] The hypothesis -d+2m+2<0 appears only inside the proof of Corollary 4.8; it should be stated in the corollary, since the claimed equivalence uses Proposition 4.4(1), which requires m ≤ (d-3)/2.
  2. [§4 (proof of Proposition 4.1(2))] In the isolated-singularities case, the proof says the local-to-global spectral sequence degenerates but does not justify why H^p(X,NK_q(X))=0 for p>0; this follows because NK_q is supported on the singular locus, and that justification should be given explicitly.
  3. [§4 (Remark 4.16)] The implication K_{-d+2m+s}-regular ⇒ m-Du Bois is stated in Remark 4.16 without repeating the affine local-complete-intersection hypotheses from Proposition 4.7; the hypotheses should be included to avoid overstatement.
  4. [Throughout] There are numerous rendering artifacts in the manuscript, including the string '/leftr⫯g⊸tl⫯ne →' in place of arrows in exact triangles and displayed equivalences, and several missing superscript braces in formulas such as z^{3m} and xyz^{m+a} in §6; these should be corrected in the final version.
Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claims require no new free parameters and no invented entities. The inputs are external theorems from Hodge theory and algebraic K-theory, listed above. The main potential circularity would be if the [CHWW1] cone characterization already contained Theorem I, but it does not: Theorem I adds a cohomological isomorphism condition and is proved from [CHSW] plus Hodge degeneration.

assumptions (9)
  • standard math Hironaka resolution: every reduced separated finite-type scheme over a characteristic-zero field admits a hyperresolution, allowing the definition of Du Bois complexes.
    Used in Section 2.1 to define Omega^p_{X/F} for general varieties.
  • standard math Cortinas-Haesemeyer-Schlichting-Weibel homotopy fibration relating cyclic homology, cdh-fibrant cyclic homology, and K-theory.
    Starting point of the proof of Theorem 7.1 in Section 7.
  • standard math Cortinas-Haesemeyer-Walker-Weibel description of NK groups via cones C^i = Cone(L^i_{X/Q} to Omega^i_{X/Q}).
    Reduces K-regularity to cohomology of cotangent and Du Bois complexes in Section 2.4.
  • standard math Hodge-to-de Rham degeneration for smooth projective complex varieties.
    Used in Proposition 7.3 to obtain the surjectivity needed for the projective characterization.
  • standard math Steenbrink vanishing: H^q Omega^p_{X/F} = 0 for p+q > dim X.
    Theorem 3.1, used throughout Sections 3 and 4.
  • domain assumption Mustata-Popa codimension bound: an m-Du Bois local complete intersection satisfies codim X_sing >= 2m+1.
    Theorem 4.12, a key input for Theorem F and Corollary 4.14.
  • domain assumption Graf and Miller-Vassiliadou results on torsion and reflexivity of Kaehler differentials for local complete intersections.
    Theorem 4.13, used to upgrade pre-Du Bois conditions to Du Bois conditions.
  • standard math Minimal exponent characterization: a local complete intersection is m-Du Bois if and only if its minimal exponent is at least m+r, where r is the embedding codimension.
    Used to derive Theorem A and Corollary 4.15 from [JKSY, MOPW, CDM].
  • standard math Base change spectral sequences in Lemma 2.10, following Kassel-Sletsjoe and Cortinas-Haesemeyer-Weibel.
    The mechanism for passing between the base field and C in Sections 2.5, 3, and 7.

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Pith. "Pith review of On higher Du Bois singularities and $K$-regularity." pith.science (2026). https://pith.science/paper/U5O6UPKE

@misc{pith2026250412402,
  author       = {Pith},
  title        = {Pith review of: On higher Du Bois singularities and $K$-regularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U5O6UPKE}},
  note         = {Machine review of arXiv:2504.12402}
}
abstract

We study the relationship between higher Du Bois singularities and $K$-regularity, a notion that measures the $\mathbb{A}^1$-invariance of the algebraic $K$-groups. Building on this relationship, we establish a strengthened form of Vorst's conjecture for local complete intersections in characteristic zero. Our work also provides tools to construct new examples that illustrate various phenomena in the study of $K$-regularity. The main inputs for our results are vanishing theorems for the Du Bois complexes.

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

  1. Negative $K$-theory and Hodge theory

    math.AG 2026-07 unverdicted novelty 7.0 of 10

    Negative K-groups of complex varieties are analyzed via mixed Hodge theory, higher singularities, Chow groups, and the Minimal Model Program.

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