Geodesic
Hodge ideals for Q-divisors: birational approach
[Idéaux de Hodge pour des Q-diviseurs : approche birationnelle]
Mustaţǎ, Mircea1 ; Popa, Mihnea2
1 Department of Mathematics, University of Michigan Ann Arbor, MI 48109, USA
2 Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208, USA
Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 283-328.

Voir la notice de l'article provenant de la source Numdam

We develop the theory of Hodge ideals for Q-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 V-filtration is analyzed in a sequel.

Nous développons la théorie des idéaux de Hodge pour les Q-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 V-filtration dans un autre article.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.94
Classification : 14F10, 14J17, 32S25, 14F17
Keywords: Hodge ideals, D-modules, Hodge filtration, vanishing theorems, minimal exponent
Mots-clés : Idéaux de Hodge, D-modules, filtration de Hodge, théorèmes d’annulation, exposant minimal

Mustaţǎ, Mircea 1 ; Popa, Mihnea 2

1 Department of Mathematics, University of Michigan Ann Arbor, MI 48109, USA
2 Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{JEP_2019__6__283_0,
     author = {Musta\c{t}ǎ, Mircea and Popa, Mihnea},
     title = {Hodge ideals for ${\protect \bf Q}$-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},
     zbl = {07070281},
     mrnumber = {3959075},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5802/jep.94/}
}
TY  - JOUR
AU  - Mustaţǎ, Mircea
AU  - Popa, Mihnea
TI  - Hodge ideals for ${\protect \bf Q}$-divisors: birational approach
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2019
SP  - 283
EP  - 328
VL  - 6
PB  - École polytechnique
UR  - http://geodesic.mathdoc.fr/articles/10.5802/jep.94/
DO  - 10.5802/jep.94
LA  - en
ID  - JEP_2019__6__283_0
ER  - 
%0 Journal Article
%A Mustaţǎ, Mircea
%A Popa, Mihnea
%T Hodge ideals for ${\protect \bf Q}$-divisors: birational approach
%J Journal de l’École polytechnique — Mathématiques
%D 2019
%P 283-328
%V 6
%I École polytechnique
%U http://geodesic.mathdoc.fr/articles/10.5802/jep.94/
%R 10.5802/jep.94
%G en
%F JEP_2019__6__283_0
Mustaţǎ, Mircea; Popa, Mihnea. Hodge ideals for ${\protect \bf Q}$-divisors: birational approach. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 283-328. doi : 10.5802/jep.94. http://geodesic.mathdoc.fr/articles/10.5802/jep.94/

[Bjö93] Björk, J.-E. Analytic 𝒟-modules and applications, Mathematics and its Applications, Kluwer Academic Publisher, Dordrecht, 1993 | MR

[Dim04] Dimca, A. Sheaves in topology, Universitext, Springer-Verlag, Berlin, New York, 2004 | DOI | Zbl

[DMST06] Dimca, A.; Maisonobe, Ph.; Saito, Morihiko; Torrelli, T. Multiplier ideals, V-filtrations and transversal sections, Math. Ann., Volume 336 (2006) no. 4, pp. 901-924 | DOI | Zbl | MR

[Dut18] Dutta, Yajnaseni Vanishing for Hodge ideals on toric varieties (2018) (arXiv:1807.11911, to appear in Math. Nachr.)

[Ful93] Fulton, W. Introduction to toric varieties, Ann. of Math. Studies, 131, Princeton University Press, Princeton, NJ, 1993 | MR

[GKKP11] Greb, Daniel; Kebekus, Stefan; Kovács, Sándor J.; Peternell, Thomas Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci., Volume 114 (2011), pp. 87-169 | Zbl | DOI | MR

[HTT08] Hotta, R.; Takeuchi, K.; Tanisaki, T. D-modules, perverse sheaves, and representation theory, Progress in Math., 236, Birkhäuser, Boston, Basel, Berlin, 2008 | Zbl | MR

[Laz04] Lazarsfeld, R. Positivity in algebraic geometry. II, Ergeb. Math. Grenzgeb. (3), 49, Springer-Verlag, Berlin, 2004 | MR | Zbl

[MP16] Mustaţă, Mircea; Popa, Mihnea Hodge ideals (2016) (arXiv:1605.08088, to appear in Memoirs of the AMS) | Zbl

[MP18a] Mustaţă, Mircea; Popa, Mihnea Hodge ideals for Q-divisors, V-filtration, and minimal exponent (2018) (arXiv:1807.01935)

[MP18b] Mustaţă, Mircea; Popa, Mihnea Restriction, subadditivity, and semicontinuity theorems for Hodge ideals, Internat. Math. Res. Notices (2018) no. 11, pp. 3587-3605 | Zbl | MR | DOI

[Pop19] Popa, Mihnea 𝒟-modules in birational geometry, Proceedings of the ICM (Rio de Janeiro, 2018), Vol. 2, World Scientific Publishing Co., Inc., River Edge, NJ, 2019, pp. 781-806

[Sai88] Saito, Morihiko Modules de Hodge polarisables, Publ. RIMS, Kyoto Univ., Volume 24 (1988) no. 6, pp. 849-995 | DOI | Zbl | MR

[Sai90] Saito, Morihiko Mixed Hodge modules, Publ. RIMS, Kyoto Univ., Volume 26 (1990) no. 2, pp. 221-333 | DOI | MR | Zbl

[Sai07] Saito, Morihiko Direct image of logarithmic complexes and infinitesimal invariants of cycles, Algebraic cycles and motives. Vol. 2 (London Math. Soc. Lecture Note Ser.), Volume 344, Cambridge Univ. Press, Cambridge, 2007, pp. 304-318 | Zbl | DOI | MR

[Sai09] Saito, Morihiko On the Hodge filtration of Hodge modules, Moscow Math. J., Volume 9 (2009) no. 1, pp. 161-191 | Zbl | MR

[Sai16] Saito, Morihiko Hodge ideals and microlocal V-filtration (2016) (arXiv:1612.08667)

[Sai17] Saito, Morihiko A young person’s guide to mixed Hodge modules, Hodge theory and L 2 -analysis (Adv. Lect. Math. (ALM)), Volume 39, Int. Press, Somerville, MA, 2017, pp. 517-553 | Zbl | MR

[Zha18] Zhang, Mingyi Hodge filtration for Q-divisors with quasi-homogeneous isolated singularities (2018) (arXiv:1810.06656)

Cité par Sources :