We develop the theory of Hodge ideals for -divisors by means of log resolutions, extending our previous work on reduced hypersurfaces. We prove local (non-)triviality criteria and a global vanishing theorem, as well as other analogues of standard results from the theory of multiplier ideals, and we derive a new local vanishing theorem. The connection with the -filtration is analyzed in a sequel.
Nous développons la théorie des idéaux de Hodge pour les -diviseurs à l’aide de log résolutions, généralisant notre précédent travail sur les hypersurfaces réduites. Nous obtenons des critères de (non) trivialité locale et un théorème d’annulation global, ainsi que d’autres analogues de résultats standard de la théorie des idéaux multiplicateurs, et nous en déduisons un nouveau théorème d’annulation local. Nous analysons la relation avec la -filtration dans un autre article.
Accepted:
Published online:
DOI: 10.5802/jep.94
Keywords: Hodge ideals, D-modules, Hodge filtration, vanishing theorems, minimal exponent
Mot clés : Idéaux de Hodge, D-modules, filtration de Hodge, théorèmes d’annulation, exposant minimal
Mircea Mustaţǎ 1; Mihnea Popa 2
@article{JEP_2019__6__283_0, author = {Mircea Musta\c{t}ǎ and Mihnea Popa}, title = {Hodge ideals for $\protect \mathbf{Q}$-divisors: birational~approach}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {283--328}, publisher = {\'Ecole polytechnique}, volume = {6}, year = {2019}, doi = {10.5802/jep.94}, mrnumber = {3959075}, zbl = {07070281}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.94/} }
TY - JOUR AU - Mircea Mustaţǎ AU - Mihnea Popa TI - Hodge ideals for $\protect \mathbf{Q}$-divisors: birational approach JO - Journal de l’École polytechnique — Mathématiques PY - 2019 SP - 283 EP - 328 VL - 6 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.94/ DO - 10.5802/jep.94 LA - en ID - JEP_2019__6__283_0 ER -
%0 Journal Article %A Mircea Mustaţǎ %A Mihnea Popa %T Hodge ideals for $\protect \mathbf{Q}$-divisors: birational approach %J Journal de l’École polytechnique — Mathématiques %D 2019 %P 283-328 %V 6 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.94/ %R 10.5802/jep.94 %G en %F JEP_2019__6__283_0
Mircea Mustaţǎ; Mihnea Popa. Hodge ideals for $\protect \mathbf{Q}$-divisors: birational approach. Journal de l’École polytechnique — Mathématiques, Volume 6 (2019), pp. 283-328. doi : 10.5802/jep.94. https://jep.centre-mersenne.org/articles/10.5802/jep.94/
[Bjö93] - Analytic -modules and applications, Mathematics and its Applications, Kluwer Academic Publisher, Dordrecht, 1993 | MR
[Dim04] - Sheaves in topology, Universitext, Springer-Verlag, Berlin, New York, 2004 | DOI | Zbl
[DMST06] - “Multiplier ideals, -filtrations and transversal sections”, Math. Ann. 336 (2006) no. 4, p. 901-924 | DOI | MR | Zbl
[Dut18] - “Vanishing for Hodge ideals on toric varieties” (2018), arXiv:1807.11911, to appear in Math. Nachr.
[Ful93] - Introduction to toric varieties, Ann. of Math. Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993 | MR
[GKKP11] - “Differential forms on log canonical spaces”, Publ. Math. Inst. Hautes Études Sci. 114 (2011), p. 87-169 | DOI | MR | Zbl
[HTT08] - -modules, perverse sheaves, and representation theory, Progress in Math., vol. 236, Birkhäuser, Boston, Basel, Berlin, 2008 | MR | Zbl
[Laz04] - Positivity in algebraic geometry. II, Ergeb. Math. Grenzgeb. (3), vol. 49, Springer-Verlag, Berlin, 2004 | MR | Zbl
[MP16] - “Hodge ideals” (2016), arXiv:1605.08088, to appear in Memoirs of the AMS | Zbl
[MP18a] - “Hodge ideals for -divisors, -filtration, and minimal exponent” (2018), arXiv:1807.01935 | Zbl
[MP18b] - “Restriction, subadditivity, and semicontinuity theorems for Hodge ideals”, Internat. Math. Res. Notices (2018) no. 11, p. 3587-3605 | DOI | MR | Zbl
[Pop19] - “-modules in birational geometry”, in Proceedings of the ICM (Rio de Janeiro, 2018), Vol. 2, World Scientific Publishing Co., Inc., River Edge, NJ, 2019, p. 781-806
[Sai88] - “Modules de Hodge polarisables”, Publ. RIMS, Kyoto Univ. 24 (1988) no. 6, p. 849-995 | DOI | MR | Zbl
[Sai90] - “Mixed Hodge modules”, Publ. RIMS, Kyoto Univ. 26 (1990) no. 2, p. 221-333 | DOI | MR | Zbl
[Sai07] - “Direct image of logarithmic complexes and infinitesimal invariants of cycles”, in Algebraic cycles and motives. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 344, Cambridge Univ. Press, Cambridge, 2007, p. 304-318 | DOI | MR | Zbl
[Sai09] - “On the Hodge filtration of Hodge modules”, Moscow Math. J. 9 (2009) no. 1, p. 161-191 | MR | Zbl
[Sai16] - “Hodge ideals and microlocal -filtration” (2016), arXiv:1612.08667 | Zbl
[Sai17] - “A young person’s guide to mixed Hodge modules”, in Hodge theory and -analysis, Adv. Lect. Math. (ALM), vol. 39, Int. Press, Somerville, MA, 2017, p. 517-553 | MR | Zbl
[Zha18] - “Hodge filtration for -divisors with quasi-homogeneous isolated singularities” (2018), arXiv:1810.06656
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