Hodge ideals and spectrum of isolated hypersurface singularities

Jung, Seung-Jo; Kim, In-Kyun; Saito, Morihiko; Yoon, Youngho

doi:10.5802/aif.3453

Hodge ideals and spectrum of isolated hypersurface singularities
[Les idéaux de Hodge et le spectre pour les singularités isolées d’hypersurfaces]

Jung, Seung-Jo ¹ ; Kim, In-Kyun ² ; Saito, Morihiko ³ ; Yoon, Youngho ⁴

¹ Department of Mathematics Education, and Institute of Pure and Applied Mathematics Jeonbuk National University Jeonju, 54896 (Korea)
² Department of mathematics, Yonsei University 50 Yonsei-Ro, Seoul 03722 (Korea)
³ RIMS Kyoto University Kyoto 606-8502 (Japan)
⁴ Department of Mathematics, Chungnam National University 99 Daehak-ro, Daejeon 34134 (Korea)

Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 465-510.

Résumé
Abstract

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 Mustață 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.

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 Mustață and Popa, these two spectra coincide in the weighted homogeneous case. We prove sufficient conditions for their coincidence and non-coincidence in some non-weighted-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 M. Zhang for the non-degenerate case, and present an example where the Hodge ideals are not weakly decreasing even modulo the Jacobian ideal.

Reçu le : 2019-11-21
Révisé le : 2020-07-12
Accepté le : 2020-11-13
Publié le : 2022-07-07

Zbl

DOI : 10.5802/aif.3453

Classification : 14F10
Keywords: Hodge ideal, spectrum, $V$-filtration
Mot clés : idéal de Hodge, spectre, $V$-filtration

Affiliations des auteurs :

Jung, Seung-Jo ¹ ; Kim, In-Kyun ² ; Saito, Morihiko ³ ; Yoon, Youngho ⁴

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Hodge ideals and spectrum of isolated hypersurface singularities},
     journal = {Annales de l'Institut Fourier},
     pages = {465--510},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
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     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3453/}
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Jung, Seung-Jo; Kim, In-Kyun; Saito, Morihiko; Yoon, Youngho. Hodge ideals and spectrum of isolated hypersurface singularities. Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 465-510. doi : 10.5802/aif.3453. https://aif.centre-mersenne.org/articles/10.5802/aif.3453/

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