We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal V -filtration modulo the Jacobian ideal. Via the Tjurina subspectrum, we can compare the Hodge ideal spectrum with the Steenbrink spectrum which can be defined by the microlocal V -filtration. As a consequence of a formula of Mustata and Popa, these two spectra coincide in the weighted homogeneous case. We prove sufficient conditions for their coincidence and non-coincidence in some nonweighted-homogeneous cases where the defining function is semi-weighted-homogeneous or with non-degenerate Newton boundary in most cases. We also show that the convenience condition can be avoided in a formula of Zhang for the non-degenerate case, and present an example where the Hodge ideals are not weakly decreasing even modulo the Jacobian ideal.

For a complex algebraic variety X, we introduce higher p-Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms Ω q X and the shifted graded pieces of the Du Bois complex Ω q X for q ≤ p. If X is a reduced hypersurface, we show that higher p-Du Bois singularity coincides with higher p-log canonical singularity, generalizing a well-known theorem for p = 0. The assertion that p-log canonicity implies p-Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently calculating the depth of the two sheaves. Our method seems much simpler using directly the acyclicity of Koszul complex in a certain range, which enables us to produce the desired isomorphisms immediately. We also improve some non-vanishing assertion shown by them, using 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.

For a complex algebraic variety 𝑋, we introduce higher 𝑝-Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms Ω 𝑞 𝑋 and the shifted graded pieces of the Du Bois complex Ω 𝑞 𝑋 for 𝑞 ⩽ 𝑝. If 𝑋 is a reduced hypersurface, we show that higher 𝑝-Du Bois singularity coincides with higher 𝑝-log canonical singularity, generalizing a well-known M S C 2 0 2 0 14B05, 14F10, 14F17, 32S35 (primary)

We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal Vfiltration modulo the Jacobian ideal. Via the Tjurina subspectrum, we can compare the Hodge ideal spectrum with the Steenbrink spectrum which can be defined by the microlocal V -filtration. As a consequence of a formula of Mustat , ă and Popa, these two spectra coincide in the weighted homogeneous case. We prove sufficient conditions for their coincidence and non-coincidence in some non-weightedhomogeneous cases where the defining function is semi-weighted-homogeneous or with non-degenerate Newton boundary in most cases. We also show that the convenience condition can be avoided in a formula of M. Zhang for the non-degenerate case, and present an example where the Hodge ideals are not weakly decreasing even modulo the Jacobian ideal.Résumé. -Nous introduisons le spectre d'idéaux de Hodge pour les singularités isolées d'hypersurfaces, qui nous permet de connaître la différence entre les idéaux de Hodge et la V -filtration microlocale modulo l'idéal jacobien. Par l'intermédiaire du sous-spectre de Tjurina, nous pouvons comparer le spectre d'idéaux de Hodge avec celui de Steenbrink qu'on peut définir en utilisant la V -filtration microlocale. Comme conséquence d'une formule de Mustat , ă et Popa, les deux spectres coïncident dans le cas de singularités isolées quasi-homogènes. Nous donnons quelques conditions suffisantes pour leur coïncidence et non-coïncidence dans quelques cas de singularités non-quasi-homogènes où les fonctions sont semi-quasi-homogènes ou non-dégénérées par rapport à leur polyèdre de Newton. Nous prouvons aussi que la condition de commodité peut être évitée dans la formule de M. Zhang dans le cas non-dégénéré, et montrons un exemple où les idéaux de Hodge ne sont pas faiblement décroissants même modulo l'idéal jacobien.

A 3-fold terminal quotient singularity X = C 3 /G admits the economic resolution Y → X, which is "close to being crepant". This paper proves that the economic resolution Y is isomorphic to a distinguished component of a moduli space of certain G-equivariant objects using the King stability condition θ introduced by K ֒ edzierski [11].