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On $k$-Du Bois and $k$-rational singularities

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arxiv 2306.03977 v2 pith:M6LWVGW2 submitted 2023-06-06 math.AG

Wanchun Shen , Sridhar Venkatesh , Anh Duc Vo This is my paper · ORCID
classification math.AG
keywords boissingularitiesrationalextendingnotionspreviousahleraspects
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abstract

We introduce new notions of $k$-Du Bois and $k$-rational singularities, extending the previous definitions in the case of local complete intersections (lci), to include natural examples outside of this setting. We study the stability of these notions under general hyperplane sections and show that varieties with $k$-rational singularities are $k$-Du Bois, extending previous results in [MP22b] and [FL22b] in the lci and the isolated singularities cases. In the process, we identify the aspects of the theory that depend only on the vanishing of higher cohomologies of Du Bois complexes (or related objects), and not on the behaviour of the K\"ahler differentials.

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Cited by 5 Pith papers

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

  1. Complexes of differential forms and singularities: The injectivity theorem

    math.AG 2025-05 accept novelty 8.0 of 10

    Kovács proves that for varieties with pre-(m-1)-Du Bois singularities, the Grothendieck dual of the m-th graded Du Bois complex injects into the dual of its zeroth cohomology sheaf on cohomology, confirming Conjecture G.

  2. Deformations, local freeness, and base change for higher Du Bois singularities

    math.AG 2026-08 conditional novelty 7.0 of 10

    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.

  3. 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...

  4. $K_2$-regularity and normality

    math.AG 2025-01 conditional novelty 7.0 of 10

    K2-regularity forces normality in all dimensions, and for affine local complete intersections K(p+1)-regularity forces regularity in codimension 2p.

  5. Reider-type theorems on normal surfaces via Bridgeland stability

    math.AG 2024-11 conditional novelty 7.0 of 10

    Using Bridgeland stability on normal surfaces, the authors prove Reider-type separation-of-jets bounds for ω_X⊗L^a, including positive characteristic and Du Bois variants, recovering optimal Fujita constants when C_X=0.

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