We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal -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 -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.
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 -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 -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.
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DOI: 10.5802/aif.3453
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
@article{AIF_2022__72_2_465_0, author = {Jung, Seung-Jo and Kim, In-Kyun and Saito, Morihiko and Yoon, Youngho}, 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}, number = {2}, year = {2022}, doi = {10.5802/aif.3453}, zbl = {07554662}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3453/} }
TY - JOUR AU - Jung, Seung-Jo AU - Kim, In-Kyun AU - Saito, Morihiko AU - Yoon, Youngho TI - Hodge ideals and spectrum of isolated hypersurface singularities JO - Annales de l'Institut Fourier PY - 2022 SP - 465 EP - 510 VL - 72 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3453/ DO - 10.5802/aif.3453 LA - en ID - AIF_2022__72_2_465_0 ER -
%0 Journal Article %A Jung, Seung-Jo %A Kim, In-Kyun %A Saito, Morihiko %A Yoon, Youngho %T Hodge ideals and spectrum of isolated hypersurface singularities %J Annales de l'Institut Fourier %D 2022 %P 465-510 %V 72 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3453/ %R 10.5802/aif.3453 %G en %F AIF_2022__72_2_465_0
Jung, Seung-Jo; Kim, In-Kyun; Saito, Morihiko; Yoon, Youngho. Hodge ideals and spectrum of isolated hypersurface singularities. Annales de l'Institut Fourier, Volume 72 (2022) no. 2, pp. 465-510. doi : 10.5802/aif.3453. https://aif.centre-mersenne.org/articles/10.5802/aif.3453/
[1] Singularity Theory I, Springer, 1998 (translated from the Russian by A. Iacob) | DOI | Zbl
[2] 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] Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscr. Math., Volume 2 (1970), pp. 103-161 | DOI | Zbl
[4] On Hodge spectrum and multiplier ideals, Math. Ann., Volume 327 (2003) no. 2, pp. 257-270 | DOI | MR | Zbl
[5] Multiplier ideals, -filtration, and spectrum, J. Algebr. Geom., Volume 14 (2005) no. 2, pp. 269-282 | DOI | MR | Zbl
[6] Singular 4-0-2 — A computer algebra system for polynomial computations, 2015 (available at http://www.singular.uni-kl.de)
[7] Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, 163, Springer, 1970 | DOI | Zbl
[8] Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J., Volume 99 (1999) no. 2, pp. 285-309 | MR | Zbl
[9] Multiplier ideals, -filtrations and transversal sections, Math. Ann., Volume 336 (2006) no. 4, pp. 901-924 | DOI | MR | Zbl
[10] A generalization of Griffiths’ theorem on rational integrals, Duke Math. J., Volume 135 (2006) no. 2, pp. 303-326 | MR | Zbl
[11] Koszul complexes and spectra of projective hypersurfaces with isolated singularities (2014) (https://arxiv.org/abs/1212.1081)
[12] 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] Jumping coefficients of multiplier ideals, Duke Math. J., Volume 123 (2004) no. 3, pp. 469-506 | MR | Zbl
[14] Local Analytic Geometry: Basic Theory and Applications, Advanced Lectures in Mathematics, Springer, 2000 | DOI | Zbl
[15] Hodge ideal and spectrum of weighted homogeneous isolated singularities (2018) (https://arxiv.org/abs/1812.07298)
[16] -functions and holonomic systems, Invent. Math., Volume 38 (1976), pp. 33-53 | DOI | Zbl
[17] 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] Polyèdres de Newton et nombres de Milnor, Invent. Math., Volume 32 (1976), pp. 1-31 | DOI | Zbl
[19] 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] 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] Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math., Volume 69 (1982), pp. 243-251 | DOI | MR | Zbl
[22] Thom–Sebastiani theorems for filtered -modules and for multiplier ideals, Int. Math. Res. Not. IMRN (2020) no. 1, pp. 91-111 | DOI | MR | Zbl
[23] Hodge ideals, Memoirs of the American Mathematical Society, 262, American Mathematical Society, 2019 | DOI | MR | Zbl
[24] Hodge ideals for -divisors: birational approach, J. Éc. polytech. Math., Volume 6 (2019), pp. 283-328 | DOI | Numdam | MR | Zbl
[25] Hodge ideals for -divisors, -filtration, and minimal exponent, Forum Math. Sigma, Volume 8 (2020), e19 | DOI | MR | Zbl
[26] Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math., Volume 14 (1971), pp. 123-142 | DOI | Zbl
[27] Exponents and Newton polyhedra of isolated hypersurface singularities, Math. Ann., Volume 281 (1988) no. 3, pp. 411-417 | DOI | MR | Zbl
[28] Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., Volume 24 (1988) no. 6, pp. 849-995 | DOI | MR | Zbl
[29] On the structure of Brieskorn lattice, Ann. Inst. Fourier, Volume 39 (1989) no. 1, pp. 27-72 | DOI | Numdam | MR | Zbl
[30] Mixed Hodge modules, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 2, pp. 221-333 | DOI | MR | Zbl
[31] On Steenbrink’s conjecture, Math. Ann., Volume 289 (1991) no. 4, pp. 703-716 | DOI | MR | Zbl
[32] Period mapping via Brieskorn modules, Bull. Soc. Math. Fr., Volume 119 (1991) no. 2, pp. 141-171 | DOI | Numdam | MR | Zbl
[33] On microlocal -function, Bull. Soc. Math. Fr., Volume 122 (1994) no. 2, pp. 163-184 | DOI | Numdam | MR | Zbl
[34] On the Hodge filtration of Hodge modules, Mosc. Math. J., Volume 9 (2009) no. 1, pp. 161-191 | MR | Zbl
[35] Deformations of abstract Brieskorn lattices (2017) (https://arxiv.org/abs/1707.07480)
[36] Hodge ideals and microlocal -filtration (2017) (https://arxiv.org/abs/1612.08667)
[37] -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] Singularities of Hypersurfaces and -Function, Proceedings of workshop in 1973 (RIMS Kokyuroku), Volume 225, RIMS, Research Institute for Mathematical Sciences, 1975
[39] On the mixed Hodge structure on the cohomology of the Milnor fibre, Math. Ann., Volume 271 (1985), pp. 641-665 | DOI | MR | Zbl
[40] Intersection form for quasi-homogeneous singularities, Compos. Math., Volume 34 (1977), pp. 211-223 | Numdam | MR | Zbl
[41] 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] 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] Asymptotic Hodge structure in the vanishing cohomology, Math. USSR, Izv., Volume 18 (1982), pp. 469-512 | Zbl
[44] The complex singularity index does not change along the stratum const, Funkts. Anal. Prilozh., Volume 16 (1982) no. 1, 96 | MR
[45] 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] 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] Hodge filtration and Hodge ideals for -divisors with weighted homogeneous isolated singularities (2019) (https://arxiv.org/abs/1810.06656)
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