no. 2
Hodge ideals and spectrum of isolated hypersurface singularities
[Les idéaux de Hodge et le spectre pour les singularités isolées d’hypersurfaces]
Jung, Seung-Jo1 ; Kim, In-Kyun2 ; Saito, Morihiko3 ; Yoon, Youngho4
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.

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.

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DOI : 10.5802/aif.3453
Classification : 14F10
Keywords: Hodge ideal, spectrum, $V$-filtration
Mot clés : idéal de Hodge, spectre, $V$-filtration
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)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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/

[1] Arnold, Vladimir I.; V., Goryunov Victor; Lyashko, O. V.; Vasil’ev, Viktor A. Singularity Theory I, Springer, 1998 (translated from the Russian by A. Iacob) | DOI | Zbl

[2] Briançon, Joël; Granger, Michel; Maisonobe, Philippe; Miniconi, M Algorithme de calcul du polynôme de Bernstein : Cas non dégénéré, Ann. Inst. Fourier, Volume 39 (1989) no. 3, pp. 553-610 | DOI | Numdam | Zbl

[3] Brieskorn, Egbert Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscr. Math., Volume 2 (1970), pp. 103-161 | DOI | Zbl

[4] Budur, Nero On Hodge spectrum and multiplier ideals, Math. Ann., Volume 327 (2003) no. 2, pp. 257-270 | DOI | MR | Zbl

[5] Budur, Nero; Saito, Morihiko Multiplier ideals, V-filtration, and spectrum, J. Algebr. Geom., Volume 14 (2005) no. 2, pp. 269-282 | DOI | MR | Zbl

[6] Decker, Wolfram; Greuel, Gert-Martin; Pfister, Gerhard; Schönemann, Hans Singular 4-0-2 — A computer algebra system for polynomial computations, 2015 (available at http://www.singular.uni-kl.de)

[7] Deligne, Pierre Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, 163, Springer, 1970 | DOI | Zbl

[8] Denef, Jan; Loeser, François Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J., Volume 99 (1999) no. 2, pp. 285-309 | MR | Zbl

[9] Dimca, Alexandru; Maisonobe, Philippe; Saito, Morihiko; Torrelli, T. Multiplier ideals, V-filtrations and transversal sections, Math. Ann., Volume 336 (2006) no. 4, pp. 901-924 | DOI | MR | Zbl

[10] Dimca, Alexandru; Saito, Morihiko A generalization of Griffiths’ theorem on rational integrals, Duke Math. J., Volume 135 (2006) no. 2, pp. 303-326 | MR | Zbl

[11] Dimca, Alexandru; Saito, Morihiko Koszul complexes and spectra of projective hypersurfaces with isolated singularities (2014) (https://arxiv.org/abs/1212.1081)

[12] Dimca, Alexandru; Saito, Morihiko Some remarks on limit mixed Hodge structures and spectrum, An. Ştiinţ. Univ. “Ovidius” Constanţa, Ser. Mat., Volume 22 (2014) no. 2, pp. 69-78 | MR | Zbl

[13] Ein, Lawrence; Lazarsfeld, Robert; Smith, Karen E.; Varolin, Dror Jumping coefficients of multiplier ideals, Duke Math. J., Volume 123 (2004) no. 3, pp. 469-506 | MR | Zbl

[14] de Jong, Theo; Pfister, Gerhard Local Analytic Geometry: Basic Theory and Applications, Advanced Lectures in Mathematics, Springer, 2000 | DOI | Zbl

[15] Jung, Seung-Jo; Kim, In-Kyun; Youngho, Yoon Hodge ideal and spectrum of weighted homogeneous isolated singularities (2018) (https://arxiv.org/abs/1812.07298)

[16] Kashiwara, Masaki B-functions and holonomic systems, Invent. Math., Volume 38 (1976), pp. 33-53 | DOI | Zbl

[17] Kashiwara, Masaki Vanishing cycle sheaves and holonomic systems of differential equations, Algebraic geometry, Proc. Jap.-Fr. Conf., Tokyo and Kyoto 1982 (Lecture Notes in Mathematics), Volume 1016, Springer, 1983, pp. 134-142 | MR | Zbl

[18] Kushnirenko, A. G. Polyèdres de Newton et nombres de Milnor, Invent. Math., Volume 32 (1976), pp. 1-31 | DOI | Zbl

[19] Malgrange, Bernard Le polynôme de Bernstein d’une singularité isolée, Fourier Integr. Oper. part. differ. Equat., Colloq. int. Nice 1974 (Lecture Notes in Mathematics), Volume 459, Springer, 1975, pp. 98-119 | Zbl

[20] Malgrange, Bernard Polynôme de Bernstein-Sato et cohomologie évanescente, Analysis and topology on singular spaces, II, III (Luminy, 1981) (Astérisque), Volume 101–102, Société Mathématique de France, 1983, pp. 243-267 | Zbl

[21] Mather, John N.; Yau, Stephen S.-T. Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math., Volume 69 (1982), pp. 243-251 | DOI | MR | Zbl

[22] Maxim, Laurentiu; Saito, Morihiko; Schürmann, Jörg Thom–Sebastiani theorems for filtered D-modules and for multiplier ideals, Int. Math. Res. Not. IMRN (2020) no. 1, pp. 91-111 | DOI | MR | Zbl

[23] Mustață, Mircea; Popa, Mihnea Hodge ideals, Memoirs of the American Mathematical Society, 262, American Mathematical Society, 2019 | DOI | MR | Zbl

[24] Mustață, Mircea; Popa, Mihnea Hodge ideals for Q-divisors: birational approach, J. Éc. polytech. Math., Volume 6 (2019), pp. 283-328 | DOI | Numdam | MR | Zbl

[25] Mustață, Mircea; Popa, Mihnea Hodge ideals for ℚ-divisors, V-filtration, and minimal exponent, Forum Math. Sigma, Volume 8 (2020), e19 | DOI | MR | Zbl

[26] Saito, Kyoji Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math., Volume 14 (1971), pp. 123-142 | DOI | Zbl

[27] Saito, Morihiko Exponents and Newton polyhedra of isolated hypersurface singularities, Math. Ann., Volume 281 (1988) no. 3, pp. 411-417 | DOI | MR | Zbl

[28] Saito, Morihiko Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., Volume 24 (1988) no. 6, pp. 849-995 | DOI | MR | Zbl

[29] Saito, Morihiko On the structure of Brieskorn lattice, Ann. Inst. Fourier, Volume 39 (1989) no. 1, pp. 27-72 | DOI | Numdam | MR | Zbl

[30] Saito, Morihiko Mixed Hodge modules, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 2, pp. 221-333 | DOI | MR | Zbl

[31] Saito, Morihiko On Steenbrink’s conjecture, Math. Ann., Volume 289 (1991) no. 4, pp. 703-716 | DOI | MR | Zbl

[32] Saito, Morihiko Period mapping via Brieskorn modules, Bull. Soc. Math. Fr., Volume 119 (1991) no. 2, pp. 141-171 | DOI | Numdam | MR | Zbl

[33] Saito, Morihiko On microlocal b-function, Bull. Soc. Math. Fr., Volume 122 (1994) no. 2, pp. 163-184 | DOI | Numdam | MR | Zbl

[34] Saito, Morihiko On the Hodge filtration of Hodge modules, Mosc. Math. J., Volume 9 (2009) no. 1, pp. 161-191 | MR | Zbl

[35] Saito, Morihiko Deformations of abstract Brieskorn lattices (2017) (https://arxiv.org/abs/1707.07480)

[36] Saito, Morihiko Hodge ideals and microlocal V-filtration (2017) (https://arxiv.org/abs/1612.08667)

[37] Saito, Morihiko D-modules generated by rational powers of holomorphic functions, Publ. Res. Inst. Math. Sci., Volume 57 (2021) no. 3-4, pp. 867-891 | DOI | MR | Zbl

[38] Sato, M. Singularities of Hypersurfaces and b-Function, Proceedings of workshop in 1973 (RIMS Kokyuroku), Volume 225, RIMS, Research Institute for Mathematical Sciences, 1975

[39] Scherk, John; Steenbrink, Joseph H. M. On the mixed Hodge structure on the cohomology of the Milnor fibre, Math. Ann., Volume 271 (1985), pp. 641-665 | DOI | MR | Zbl

[40] Steenbrink, Joseph H. M. Intersection form for quasi-homogeneous singularities, Compos. Math., Volume 34 (1977), pp. 211-223 | Numdam | MR | Zbl

[41] Steenbrink, Joseph H. M. Mixed Hodge structure on the vanishing cohomology, in Real and complex singularities, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), 1977, pp. 525-563 | Zbl

[42] Steenbrink, Joseph H. M. The spectrum of hypersurface singularities, Théorie de Hodge, Actes Colloq., Luminy/Fr. 1987 (Astérisque), Volume 179-180, Société Mathématique de France, 1989, pp. 163-184 | Numdam | Zbl

[43] Varchenko, Alexander N. Asymptotic Hodge structure in the vanishing cohomology, Math. USSR, Izv., Volume 18 (1982), pp. 469-512 | Zbl

[44] Varchenko, Alexander N. The complex singularity index does not change along the stratum μ=const, Funkts. Anal. Prilozh., Volume 16 (1982) no. 1, 96 | MR

[45] Varchenko, Alexander N. A lower bound for the codimension of the strata mu=const in terms of the mixed Hodge structure, Vestn. Mosk. Univ., Volume 1982 (1982) no. 6, pp. 28-31 | Zbl

[46] Varchenko, Alexander N.; Khovanskiĭ, Askold G. Asymptotic behavior of integrals over vanishing cycles and the Newton polyhedron, Dokl. Akad. Nauk SSSR, Volume 283 (1985) no. 3, pp. 521-525 | MR

[47] Zhang, Mingyi Hodge filtration and Hodge ideals for ℚ-divisors with weighted homogeneous isolated singularities (2019) (https://arxiv.org/abs/1810.06656)

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