Complexes of differential forms and singularities: The injectivity theorem

S\'andor Kov\'acs

Complexes of differential forms and singularities: The injectivity theorem

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Pith's one-line read Injective duality holds for all pre-(m-1)-Du Bois varieties.

desk verdict Kov\u00e1cs proves Conjecture G of Popa\u2013Shen\u2013Vo for arbitrary varieties with pre-(m-1)-Du Bois singularities, removing the lci/isolated restrictions; the proof is intricate but appears sound. read the letter →

arxiv 2505.09912 v5 pith:QISVJMSW submitted 2025-05-15 math.AG

Sándor Kovács This is my paper · ORCID

classification math.AG MSC 14B0514J17

keywords DuBoissingularitieshigherDeligne-DucomplexGrothendieckdualityinjectivitytheoremhyperfiltrationslocalcohomologyrational

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a local injectivity theorem for singular complex varieties: if a variety $U$ has pre-$(m-1)$-Du Bois singularities, then for every $q$ and every $p\le m$ the natural cohomology map $h^q(D_U(\Omega^p_U)) \to h^q(D_U(e\Omega^p_U))$ is injective, where $e\Omega^p_U=h^0(\Omega^p_U)$ is the zeroth cohomology sheaf of the $p$-th graded Du Bois complex. This confirms Conjecture G of Popa, Shen, and Vo, which had previously been established only for local complete intersections and for isolated singularities. The statement is purely local, about injectivity of sheaves, even though the proof has to pass through global surjectivity coming from degeneration of the Frolicher spectral sequence. The result matters because this injectivity underlies vanishing theorems, local cohomology surjectivity, and splitting criteria that feed into birational geometry and moduli theory.

What carries the argument

The argument is carried by hyper(co)filtrations of the filtered Deligne-Du Bois complex, a derived-category formalism that lets filtrations pass through arbitrary functors. A key auxiliary object, which the paper nicknames `l'eminence grise', is a complex $G^p_X(L^{-j})$ built as a mapping cone interpolating between the $h^0$-complex $e\Omega^p_X$ and the full graded Du Bois complex $\Omega^p_X$; it converts the global surjectivity supplied by degeneration of the Hodge-to-de Rham spectral sequence into hypercohomology surjections. Serre vanishing then turns these global surjectivities into the desired local injectivity, and an induction on dimension using general hyperplane sections together with a Nakayama argument (an element $f$ annihilates the kernel) closes the proof.

What would settle it

A direct refutation would be a specific variety $U$ with pre-$(m-1)$-Du Bois singularities and values $q,p\le m$ for which $h^q(D_U(\Omega^p_U)) \to h^q(D_U(e\Omega^p_U))$ has a nonzero kernel. A less direct check: exhibit a pre-$m$-Du Bois $X$ whose general hyperplane section is not pre-$m$-Du Bois, which would invalidate Proposition 5.10 and the proof's induction step.

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Extended reading notes

Core claim

The central claim is Theorem 9.1, which the paper identifies as [PSV24, Conjecture G]. Let $U$ be a variety of pure dimension $n$ with pre-$(m-1)$-Du Bois singularities. Then for each $q$ and each $p\le m$, the natural morphism $h^q(D_U(\Omega^p_U)) \to h^q(D_U(e\Omega^p_U))$ is injective, where $D_U(-)=\mathrm{RHom}_U(-,\omega_U^{\bullet})[-n]$ is the shifted Grothendieck duality functor and $e\Omega^p_U=h^0(\Omega^p_U)$. Equivalently, the natural map $\mathrm{RHom}_U(\Omega^m_U,\omega_U^{\bullet}) \to \mathrm{RHom}_U(h^0(\Omega^m_U),\omega_U^{\bullet})$ is injective on cohomology. The theorem removes the earlier lci and isolated-singularity restrictions and does not assume properness of $U$.

Load-bearing premise

The dimension induction in Theorem 9.1 relies on Proposition 5.10, which says a general hyperplane section of a variety with pre-$m$-Du Bois singularities again has pre-$m$-Du Bois singularities; if that preservation failed in a non-lci, non-isolated example, the induction would collapse.

Editorial extensions

If this is right

Conjecture H of [PSV24] follows: if $X$ has pre-$(m-1)$-Du Bois singularities and pre-$m$-Du Bois singularities away from a closed subset of dimension $r$, then $h^i(\Omega^m_X)=0$ for $0<i<\operatorname{depth} h^0(\Omega^m_X)-r-1$.
Local cohomology surjectivity holds: at a point $x\in X$ where $X$ is pre-$(m-1)$-Du Bois, the natural map $H^q_x(X,h^0(\Omega^p_X)) \to H^q_x(X,\Omega^p_X)$ is surjective for every $q$ and $p\le m$.
The theorem implies the depth inequality $\operatorname{depth} \Omega^m_X \ge \operatorname{depth} h^0(\Omega^m_X)$ for varieties with pre-$(m-1)$-Du Bois singularities.
A splitting criterion follows: if the natural morphism $h^0(\Omega^p_X)\to\Omega^p_X$ has a left inverse for each $p\le m$, then $X$ has pre-$m$-Du Bois singularities.
Pre-$m$-rational, $m$-rational, and strict $m$-rational singularities imply respectively weakly-$m$-Du Bois, $m$-Du Bois, and strict $m$-Du Bois singularities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The cone object at the heart of the proof may transfer to other duality setups, such as relative dualizing complexes for families, which could give deformation-invariance statements for higher Du Bois singularities that the paper does not state.
A natural test is whether the injectivity remains true with $h^k(\Omega^p_U)$ in place of $h^0(\Omega^p_U)$ for $k>0$; the paper's arguments only target the zeroth cohomology sheaf.
Because the proof relies on characteristic-zero Hodge degeneration, a meaningful extension would be to formulate the same conjecture for the de Rham-Witt or l-adic intersection complexes in positive or mixed characteristic.
The result suggests that `pre-$m$-Du Bois' is the natural generality for injectivity: the extra codimension and $S_2$ conditions appearing in the definition of $m$-Du Bois singularities are not needed for this particular statement.

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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here is my take: the paper proves the injectivity theorem (Conjecture G of [PSV24]) for arbitrary varieties with pre-(m-1)-Du Bois singularities, with no lci or isolated assumption. That is a genuine advance. The new technical ingredient is the "\u00e9minence grise" object in Section 8, which converts global surjectivity from Hodge theory into the local injectivity you want. Along the way it also gives Conjecture H, a local cohomology surjectivity statement, and splitting criteria for pre-m-Du Bois singularities.

What is good: the theorem is new, the main steps are accounted for, and the text is careful. There is some overlap with [SVV] on definitions and corollaries, but the approach and the central theorem are different. The author is also honest about the history: he acknowledges an earlier version had a subtle mistake, now corrected. That is a point in favor of the paper's reliability, not against it.

Where are the soft spots? The proof is long and intricate, relying on spectral sequence degenerations, 4-lemma/5-lemma chases, and a delicate proper-to-local reduction. I did not machine-check any of it, and the reader did not either. The single most load-bearing step is Proposition 5.10, which says pre-(m-1)-Du Bois persists under general hyperplane section. The stress-test note checked this and concluded it is valid, provided one applies it to the affine variety U rather than the projective closure X. I agree. If that proposition failed, the induction in Theorem 9.1 would collapse; I do not see it failing.

Minor comments: the relationship between the various definitions (pre-m-Du Bois, weakly-m-Du Bois, and the corresponding notions in [SVV]) is a little tangled, though not a mathematical problem. There is a light reference to a Wikipedia page for "\u00e9minence grise," which is unusual but harmless.

Overall: solid, significant within-field work. It deserves a serious referee and, in my view, acceptance after careful reading. I would bring it to the reading group and would cite it if I worked in the area.

Recommendation: send to peer review, with a referee who can carefully check Proposition 5.10 and the diagram in Theorem 8.1. If those hold up, the paper is correct.

Referee Report

0 major / 4 minor

Summary. The paper proves Theorem 9.1 (Conjecture G of [PSV24]): if U is a variety of pure dimension n with pre-(m-1)-Du Bois singularities, then for every q and every p ≤ m the natural morphism h^q(D_U(Ω^p_U)) → h^q(D_U(eΩ^p_U)) is injective; equivalently, RHom_U(Ω^m_U, ω_U^•) → RHom_U(h^0(Ω^m_U), ω_U^•) is injective on cohomology. The proof develops a substantial machinery of (co-)hyperfiltrations and hyperfiltered connections, establishes preservation of pre-m-Du Bois singularities under general hyperplane sections (Proposition 5.10), uses cyclic covers to obtain a global surjectivity statement (Theorem 8.1, the so-called l'éminence grise), and then converts this global surjectivity into local injectivity via Serre vanishing and a Nakayama argument. Section 10 derives applications, including [PSV24, Conjecture H], a local cohomology surjectivity statement (Theorem 10.3), splitting criteria for pre-m-Du Bois singularities (Theorems 10.4 and 10.5), and implications from higher rational to higher Du Bois singularities (Corollaries 10.11–10.13).

Significance. If correct, the main theorem settles Conjecture G of [PSV24] and, through the arguments indicated in Section 10, also Conjecture H and the intersection-Du-Bois variants. The result is significant because it removes the lci and isolated-singularity assumptions that limited earlier injectivity theorems, and it introduces a new technical apparatus—co-hyperfiltrations, the 'éminence grise' object, and the proper-to-local reduction—that is likely to be useful beyond this paper. The proof is detailed and essentially self-contained: the central chain of implications is accounted for, including the induction through general hyperplane sections, the cyclic-cover comparisons, and the Serre-vanishing/localization step. A particular strength is that the paper is explicit about the provenance of the conjecture and about its overlap with [SVV]. The main risk is the sheer intricacy of the diagram chases; I found no internal inconsistency in the arguments as written.

minor comments (4)

[Section 9, proof of Theorem 9.1] In the first paragraph of the proof, the sentence 'Then V:=U∩H has pre-(m-1)-Du Bois singularities by Proposition 5.10' is potentially misleading, because Proposition 5.10 as stated requires the ambient variety itself to have pre-(m-1)-Du Bois singularities, and the projective closure X is not assumed to have them. The intended and valid reading is that Proposition 5.10 is applied to the quasi-projective variety U with H∩U a general member of the restricted basepoint-free linear system; please rephrase the sentence to state this explicitly.
[Section 8, proof of Theorem 8.1] The notation in the hypercohomology diagrams is inconsistent: the object G^p_{X,H}(L^{-j}) already contains the twist L^{-j}, yet the displayed cohomology groups write 'G^p_{X,H}(L^{-j})⊗L^{-j}' and 'G^p_X(L^{-j})⊗L^{-j}'. The same proof also uses ν^{p,j}_X both for the morphism G^p_X(L^{-j})→Ω^p_X⊗L^{-j} and for the composition eΩ^p_X⊗L^{-j}→Ω^p_X⊗L^{-j}; this makes it hard to see that cokerβ in (9.1.13) is the cokernel of the latter. Please disambiguate the notation.
[Section 5, Proposition 5.10] The assertion that h^i(Ω^p_X ⊗^L O_H) ≃ h^i(Ω^p_X) ⊗^L O_H is justified only by reference to Corollary 2.4 and Lemma 2.6. Since those results give Tor-independence of each cohomology sheaf with O_H, one uses the hyperderived spectral sequence with E^2_{r,s} = Tor_r(h^s(Ω^p_X), O_H); stating this explicitly would improve readability of an already intricate induction.
[Throughout] There are several typographical slips: 'defintion' in Remark 4.6, 'Sung Gi park' in the Acknowledgments should be capitalized, and 'Fianlly' appears at the end of Section 9. None of these affect the mathematics, but they should be corrected in a final revision.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: Theorem 9.1 is proved from independent Hodge-theoretic input and auxiliary cone constructions, with only routine self-citations.

full rationale

The main theorem is not obtained by re-stating its hypothesis. The hypothesis 'pre-(m-1)-Du Bois' means eΩ^p_U ≃ Ω^p_U for p≤m−1, while the conclusion concerns injectivity of h^q(D_U(Ω^p_U))→h^q(D_U(eΩ^p_U)) for p≤m; the p=m case is not part of the definition. In the proof of Theorem 9.1, Proposition 5.10 is used to pass the hypothesis to V=U∩H; that proposition is proved internally by induction on p using the hyperderived spectral sequence, Tor-independence from Corollary 2.4/Lemma 2.6, and the distinguished triangles of Lemma 5.5. It does not invoke the injectivity theorem. The dimensional induction is a genuine strong induction on dim U, and the 4-lemma transfer is legitimate because the outside maps are isomorphisms from the definition of pre-(m−1)-Du Bois and the middle map is injective by the induction hypothesis. The auxiliary objects G^p_X(L^{-j}) and G^p_{X,H}(L^{-j}) in Theorem 8.1 are cones of natural filtration maps; they are constructed, not fitted to the target injectivity, and the key surjectivity (Theorem 8.1(v)) is derived from Deligne's degeneration of the Hodge-to-de Rham spectral sequence, Proposition 2.25, and Corollary 7.6, none of which is equivalent to Conjecture G. The citations to [SVV] are for definitions and for auxiliary identifications that do not enter the main induction; the citations to [Kov05] supply a general spectral sequence formalism, and Proposition 2.25 is proved in this paper. No parameter is fitted to any subset of the conclusion, and no prediction is renamed from an input. The delicate step Proposition 5.10 is valid in this setting because H is chosen general and the statement is applied to U, not to the projective closure X. Thus the derivation is self-contained; the score reflects only the presence of the author's own earlier machinery, which is not circularly load-bearing.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests on the standard machinery of Deligne-Du Bois complexes, Deligne's degeneration theorem, Grothendieck duality, and the author's prior hyperfiltration spectral sequence formalism. There are no fitted free parameters and no newly postulated entities; the auxiliary 'éminence grise' objects are explicitly constructed as cones from existing data.

assumptions (8)

domain assumption The filtered Deligne-Du Bois complex Ω^•_X exists and satisfies the stated properties (filtered resolution of the constant sheaf, torsion-free h^0 pieces).
Invoked in Sections 3.D to 3.L and throughout the paper; established by Du Bois [DB81] and Deligne [Del74], not proved here.
domain assumption For proper X, the Hodge-to-de Rham spectral sequence of a pair (X,Σ) degenerates at E1 and abuts to the Hodge filtration of the mixed Hodge structure on H^q_c(X\Σ,C).
Used in Theorem 3.10 and Lemma 7.1 to obtain the filtration F^p H^q_c(V,C) and the surjectivity in Corollary 7.4; this is Deligne's theorem.
domain assumption Resolution of singularities and existence of cubical hyperresolutions for reduced schemes of finite type over C.
Needed to define the Deligne-Du Bois complex via Rε_*Ω^•_{X_•} (Section 3.D) and to form log resolutions in Sections 5, 6, and 8.
standard math Grothendieck duality and Serre duality for proper morphisms (Lemma 2.8, Lemma 2.9).
Used to relate D_X to dualizing complexes and to identify H^j(X,A)∨ with H^{n-j}(X,D_X(A)).
standard math Serre vanishing for very ample line bundles on projective varieties.
Used around (9.1.4) to make the conjugate spectral sequence comparison an isomorphism and to handle the support of the kernel K^q.
standard math GAGA principle identifying algebraic and analytic cohomology for proper schemes.
Used in Lemma 7.1 to pass from algebraic hypercohomology to the analytic Hodge filtration.
standard math Bertini's second theorem and the behavior of general hyperplane sections with respect to singular loci.
Used in Section 5 to prove Lemma 5.2 and Proposition 5.10, which drive the dimension induction in Theorem 9.1.
domain assumption The E1 spectral sequence for hyperfiltrations from [Kov05, Thm 1.2.2].
Proposition 2.25 extends this; the global surjectivity in Theorem 8.1(v) relies on it. This is the author's own prior published tool, not a consequence of the conjecture.

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Pith. "Pith review of Complexes of differential forms and singularities: The injectivity theorem." pith.science (2026). https://pith.science/paper/QISVJMSW

@misc{pith2026250509912,
  author       = {Pith},
  title        = {Pith review of: Complexes of differential forms and singularities: The injectivity theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QISVJMSW}},
  note         = {Machine review of arXiv:2505.09912}
}

read the original abstract

In this paper, it is proved, that for varieties with (m-1)-Du Bois singularities, the natural morphism from the Grothendieck dual of the m-th graded Du Bois complex to the Grothendieck dual of its zero-th cohomology sheaf is injective on cohomology. This confirms Conjecture G of Popa, Shen, and Vo [PSV24].

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On higher Du Bois singularities and $K$-regularity
math.AG 2025-04 conditional novelty 7.0 of 10

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 characterizati...

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Works this paper leans on

54 extracted references · 51 canonical work pages · cited by 1 Pith paper

[1]

Artin : On isolated rational singularities of surfaces, Amer

M. Artin : On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129--136

work page 1966
[2]

Bruns and J

W. Bruns and J. Herzog : Cohen- M acau\-lay rings , Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993

work page 1993
[3]

J. A. Carlson : Polyhedral resolutions of algebraic varieties, Trans. Amer. Math. Soc. 292 (1985), 2, 595--612

work page 1985
[4]

Cumino, S

C. Cumino, S. Greco, and M. Manaresi : Weakly normal algebraic varieties and B ertini theorem , Proceedings of the G eometry C onference ( M ilan and G argnano, 1987), vol. 57, 1987, pp. 135--148

work page 1987
[5]

Cumino, S

C. Cumino, S. Greco, and M. Manaresi : An axiomatic approach to the second theorem of B ertini , J. Algebra 98 (1986), 1, 171--182

work page 1986
[6]

Deligne : Th\'eorie de H odge

P. Deligne : Th\'eorie de H odge. I , Actes du C ongr\`es I nternational des M ath\'ematiciens ( N ice, 1970), T ome 1, Gauthier-Villars, Paris, 1971, pp. 425--430

work page 1970
[7]

Deligne : Th\'eorie de H odge

P. Deligne : Th\'eorie de H odge. II , Inst. Hautes \'Etudes Sci. Publ. Math. (1971), 40, 5--57

work page 1971
[8]

Deligne : Th\'eorie de H odge

P. Deligne : Th\'eorie de H odge. III , Inst. Hautes \'Etudes Sci. Publ. Math. (1974), 44, 5--77

work page 1974

Show all 54 references

[9]

D u B ois : Complexe de de R ham f i ltr\'e d'une vari\'et\'e singuli\`ere , Bull

Ph . D u B ois : Complexe de de R ham f i ltr\'e d'une vari\'et\'e singuli\`ere , Bull. Soc. Math. France 109 (1981), 1, 41--81

1981
[10]

Elkik : Singularit\'es rationnelles et d\'eformations, Invent

R. Elkik : Singularit\'es rationnelles et d\'eformations, Invent. Math. 47 (1978), 2, 139--147

1978
[11]

Esnault and E

H. Esnault and E. Viehweg : Rev\^etements cycliques, Algebraic threefolds (Varenna, 1981), Lecture Notes in Math., vol. 947, Springer, Berlin, 1982, pp. 241--250

1981
[12]

Esnault and E

H. Esnault and E. Viehweg : Lectures on vanishing theorems, DMV Seminar, vol. 20, Birkh\"auser, 1992

1992
[13]

Flenner, L

H. Flenner, L. O'Carroll, and W. Vogel : Joins and intersections, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999

1999
[14]

Fran c ois Leclerc du Tremblay : https://en.wikipedia.org/wiki/Eminence_grise https://en.wikipedia.org/wiki/ \'E minence\_grise
[15]

Friedman and R

R. Friedman and R. Laza : The higher D u B ois and higher rational properties for isolated singularities , J. Algebraic Geom. 33 (2024), 3, 493--520

2024
[16]

Friedman and R

R. Friedman and R. Laza : Higher D u B ois and higher rational singularities , Duke Math. J. 173 (2024), 10, 1839--1881, Appendix by Morihiko Saito

2024
[17]

Greb : Rational singularities and quotients by holomorphic group actions, Ann

D. Greb : Rational singularities and quotients by holomorphic group actions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), 2, 413--426

2011
[18]

Grothendieck : \' E l\'ements de g\'eom\'etrie alg\'ebrique, IV

A. Grothendieck : \' E l\'ements de g\'eom\'etrie alg\'ebrique, IV . \' E tude locale des sch\'emas et des morphismes de sch\'emas. III , Inst. Hautes \'Etudes Sci. Publ. Math. (1966), 28, 255

1966
[19]

Guill \'e n, V

F. Guill \'e n, V. Navarro \ Aznar, P. Pascual \ Gainza, and F. Puerta : Hyperr\'esolutions cubiques et descente cohomologique, Lecture Notes in Mathematics, vol. 1335, Springer-Verlag, Berlin, 1988, Papers from the Seminar on Hodge-Deligne Theory held in Barcelona, 1982

1988
[20]

Hartshorne : Residues and duality, Lecture notes of a seminar on the work of A

R. Hartshorne : Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, 1966

1963
[21]

Hartshorne : Algebraic geometry, Springer-Verlag, 1977, Graduate Texts in Mathematics, No

R. Hartshorne : Algebraic geometry, Springer-Verlag, 1977, Graduate Texts in Mathematics, No. 52

1977
[22]

Huber and C

A. Huber and C. J\"order : Differential forms in the h-topology, Algebr. Geom. 1 (2014), 4, 449--478

2014
[23]

Jung, I.-K

S.-J. Jung, I.-K. Kim, M. Saito, and Y. Yoon : Higher D u B ois singularities of hypersurfaces , Proc. Lond. Math. Soc. (3) 125 (2022), 3, 543--567

2022
[24]

Kebekus and C

S. Kebekus and C. Schnell : Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities, J. Amer. Math. Soc. 34 (2021), 2, 315--368

2021
[25]

Kim : Trace for the D u B ois complex , arXiv:2507.07350

H. Kim : Trace for the D u B ois complex , arXiv:2507.07350
[26]

Koll \'a r : Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol

J. Koll \'a r : Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, 2013, with the collaboration of S \'a ndor J Kov \'a cs

2013
[27]

Koll\' a r : Families of varieties of general type, Cambridge Tracts in Mathematics, vol

J. Koll\' a r : Families of varieties of general type, Cambridge Tracts in Mathematics, vol. 231, Cambridge University Press, Cambridge, 2023, With the collaboration of Klaus Altmann and S \'a ndor Kov \'a cs

2023
[28]

Koll \'a r and S

J. Koll \'a r and S. J. Kov \'a cs : Log canonical singularities are D u B ois , J. Amer. Math. Soc. 23 (2010), 3, 791--813

2010
[29]

S. J. Kov \'a cs : Relative de R ham complex for non-smooth morphisms , Birational algebraic geometry (Baltimore, MD, 1996), Contemp. Math., vol. 207, Amer. Math. Soc., Providence, RI, 1997, pp. 89--100

1996
[30]

S. J. Kov \'a cs : Rational, log canonical, D u B ois singularities: on the conjectures of K oll\'ar and S teenbrink , Compositio Math. 118 (1999), 2, 123--133

1999
[31]

S. J. Kov \'a cs : A characterization of rational singularities, Duke Math. J. 102 (2000), 2, 187--191

2000
[32]

S. J. Kov \'a cs : Spectral sequences associated to morphisms of locally free sheaves, Recent progress in arithmetic and algebraic geometry, Contemp. Math., vol. 386, Amer. Math. Soc., Providence, RI, 2005, pp. 57--85

2005
[33]

S. J. Kov \'a cs : Du B ois pairs and vanishing theorems , Kyoto J. Math. 51 (2011), 1, 47--69

2011
[34]

S. J. Kov \'a cs , in preparation
[35]

S. J. Kov\'acs and K. Schwede : Du B ois singularities deform , Minimal models and extremal rays ( K yoto, 2011), Adv. Stud. Pure Math., vol. 70, Math. Soc. Japan, [Tokyo], 2016, pp. 49--65

2011
[36]

S. J. Kov \'a cs and K. Schwede : Inversion of adjunction for rational and D u B ois pairs , Algebra Number Theory 10 (2016), 5, 969--1000

2016
[37]

S. J. Kov \'a cs and K. E. Schwede : Hodge theory meets the minimal model program: a survey of log canonical and D u B ois singularities , Topology of stratified spaces, Math. Sci. Res. Inst. Publ., vol. 58, Cambridge Univ. Press, Cambridge, 2011, pp. 51--94

2011
[38]

S. J. Kov\' a cs and B. Taji : Hodge sheaves underlying flat projective families, Math. Z. 303 (2023), 3, Paper No. 75, 34

2023
[39]

Musta t a , S

M. Musta t a , S. Olano, M. Popa, and J. Witaszek : The D u B ois complex of a hypersurface and the minimal exponent , Duke Math. J. 172 (2023), 7, 1411--1436

2023
[40]

Musta t a and M

M. Musta t a and M. Popa : Hodge filtration on local cohomology, D u B ois complex and local cohomological dimension , Forum Math. Pi 10 (2022), Paper No. e22, 58

2022
[41]

Musta t a and M

M. Musta t a and M. Popa : On k -rational and k - D u B ois local complete intersections , Algebr. Geom. 12 (2025), 2, 237--261

2025
[42]

Namikawa : Projectivity criterion of M oishezon spaces and density of projective symplectic varieties , Internat

Y. Namikawa : Projectivity criterion of M oishezon spaces and density of projective symplectic varieties , Internat. J. Math. 13 (2002), 2, 125--135

2002
[43]

S. G. Park and M. Popa : Lefschetz theorems, Q -factoriality, and H odge symmetry for singular varieties
[44]

C. A. M. Peters and J. H. M. Steenbrink : Mixed H odge structures , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 52, Springer-Verlag, 2008

2008
[45]

M. Popa, W. Shen, and A. D. Vo : Injectivity and vanishing for the D u B ois complexes of isolated singularities , arXiv:2409.18019

arXiv
[46]

Saito : Mixed H odge complexes on algebraic varieties , Math

M. Saito : Mixed H odge complexes on algebraic varieties , Math. Ann. 316 (2000), 2, 283--331

2000
[47]

Serre : G\'eom\'etrie alg\'ebrique et g\'eom\'etrie analytique, Ann

J.-P. Serre : G\'eom\'etrie alg\'ebrique et g\'eom\'etrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955-56), 1--42

1955
[48]

W. Shen, S. G. Venkatesh, and A. D. Vo : On k- D u B ois and k-rational singularities , arXiv:2306.03977

arXiv
[49]

J. H. M. Steenbrink : Mixed H odge structures associated with isolated singularities , Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., 1983, pp. 513--536

1981
[50]

J. H. M. Steenbrink : Vanishing theorems on singular spaces, Ast\'erisque (1985), 130, 330--341, Differential systems and singularities (Luminy, 1983)

1985
[51]

Tighe : The holomorphic extension property for k- D u B ois singularities , arXiv.2312.01245

B. Tighe : The holomorphic extension property for k- D u B ois singularities , arXiv.2312.01245

arXiv
[52]

Viehweg : Vanishing theorems, J

E. Viehweg : Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1--8

1982
[53]

Voisin : Hodge theory and complex algebraic geometry

C. Voisin : Hodge theory and complex algebraic geometry. I , E nglish ed., Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2007

2007
[54]

Stacks Project Authors : Stacks P roject , http://stacks.math.columbia.edu

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Artin : On isolated rational singularities of surfaces, Amer

M. Artin : On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129--136

work page 1966

[2] [2]

Bruns and J

W. Bruns and J. Herzog : Cohen- M acau\-lay rings , Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993

work page 1993

[3] [3]

J. A. Carlson : Polyhedral resolutions of algebraic varieties, Trans. Amer. Math. Soc. 292 (1985), 2, 595--612

work page 1985

[4] [4]

Cumino, S

C. Cumino, S. Greco, and M. Manaresi : Weakly normal algebraic varieties and B ertini theorem , Proceedings of the G eometry C onference ( M ilan and G argnano, 1987), vol. 57, 1987, pp. 135--148

work page 1987

[5] [5]

Cumino, S

C. Cumino, S. Greco, and M. Manaresi : An axiomatic approach to the second theorem of B ertini , J. Algebra 98 (1986), 1, 171--182

work page 1986

[6] [6]

Deligne : Th\'eorie de H odge

P. Deligne : Th\'eorie de H odge. I , Actes du C ongr\`es I nternational des M ath\'ematiciens ( N ice, 1970), T ome 1, Gauthier-Villars, Paris, 1971, pp. 425--430

work page 1970

[7] [7]

Deligne : Th\'eorie de H odge

P. Deligne : Th\'eorie de H odge. II , Inst. Hautes \'Etudes Sci. Publ. Math. (1971), 40, 5--57

work page 1971

[8] [8]

Deligne : Th\'eorie de H odge

P. Deligne : Th\'eorie de H odge. III , Inst. Hautes \'Etudes Sci. Publ. Math. (1974), 44, 5--77

work page 1974