Higher Du Bois singularities of hypersurfaces

By Seung-Jo Jung, In-Kyun Kim, Morihiko Saito, Youngho Yoon. Proceedings of the London Mathematical Society, 2021

54 citations recorded by the source corpus.

For a complex algebraic variety X$X$ , we introduce higher p$p$ ‐Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms ΩXq$\Omega _X^q$ and the shifted graded pieces of the Du Bois complex Ω̲Xq$\underline{\Omega }_X^q$ for q⩽p$q\leqslant p$ . If X$X$ is a reduced hypersurface, we show that higher p$p$ ‐Du Bois singularity coincides with higher p$p$ ‐log canonical singularity, generalizing a well‐known theorem for p=0$p=0$ . The assertion that p$p$ ‐log canonicity implies p$p$ ‐Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently as a corollary of two theorems asserting that the sheaves of reflexive differential forms ΩX[q]$\Omega _X^{[q]}$ ( q⩽p$q\leqslant p$ ) coincide with ΩXq$\Omega _X^q$ and Ω̲Xq$\underline{\Omega }_X^q$ , respectively, and these are shown by calculating the depth of the latter two sheaves. We construct explicit isomorphisms between ΩXq$\Omega _X^q$ and Ω̲Xq$\underline{\Omega }_X^q$ applying the acyclicity of a Koszul complex in a certain range. We also improve some non‐vanishing assertion shown by them using mixed Hodge modules and the Tjurina subspectrum in the isolated singularity case. This is useful for instance to estimate the lower bound of the maximal root of the reduced Bernstein–Sato polynomial in the case where a quotient singularity is a hypersurface and its singular locus has codimension at most 4.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f92ad6b2ab054983b26d10720437f8fcab4dca82

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension (2021, background): “Following the terminology from [24], we say that Z has only higher p-Du Bois singularities if the canonical morphisms ΩZ → Ω k Z are isomorphisms for all 0 ≤ k ≤ p.”