Multiplier ideals, $V$-filtration, and spectrum
Authors:
Nero Budur and Morihiko Saito
Journal:
J. Algebraic Geom. 14 (2005), 269-282
DOI:
https://doi.org/10.1090/S1056-3911-04-00398-4
Published electronically:
December 9, 2004
MathSciNet review:
2123230
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Abstract: For an effective divisor on a smooth algebraic variety or a complex manifold, we show that the associated multiplier ideals coincide essentially with the filtration induced by the filtration $V$ constructed by B. Malgrange and M. Kashiwara. This implies another proof of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith, and D. Varolin that any jumping coefficient in the interval $(0,1]$ is a root of the Bernstein-Sato polynomial up to sign. We also give a refinement (using mixed Hodge modules) of the formula for the coefficients of the spectrum for exponents not greater than one or greater than the dimension of the variety minus one.
- A. A. BeÄlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) AstĂ©risque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5â171 (French). MR 751966
- A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic $D$-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR 882000
3 Budur, N., thesis.
- Nero Budur, On Hodge spectrum and multiplier ideals, Math. Ann. 327 (2003), no. 2, 257â270. MR 2015069, DOI https://doi.org/10.1007/s00208-003-0450-9
5 ---, Multiplier ideals and filtered $D$-modules, preprint.
- Pierre Deligne, ThĂ©orie de Hodge. I, Actes du CongrĂšs International des MathĂ©maticiens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 425â430 (French). MR 0441965
7 ---, Le formalisme des cycles Ă©vanescents, in SGA7 XIII and XIV, Lect. Notes in Math. 340, Springer, Berlin, 1973, pp. 82â115 and 116â164.
- Jean-Pierre Demailly, $L^2$-methods and effective results in algebraic geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (ZĂŒrich, 1994) BirkhĂ€user, Basel, 1995, pp. 817â827. MR 1403982
- Jan Denef and François Loeser, Motivic Igusa zeta functions, J. Algebraic Geom. 7 (1998), no. 3, 505â537. MR 1618144
- Lawrence Ein, Multiplier ideals, vanishing theorems and applications, Algebraic geometryâSanta Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 203â219. MR 1492524, DOI https://doi.org/10.1090/s0894-0347-97-00223-3
- Lawrence Ein and Robert Lazarsfeld, Singularities of theta divisors and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997), no. 1, 243â258. MR 1396893, DOI https://doi.org/10.1090/S0894-0347-97-00223-3
- Lawrence Ein, Robert Lazarsfeld, Karen E. Smith, and Dror Varolin, Jumping coefficients of multiplier ideals, Duke Math. J. 123 (2004), no. 3, 469â506. MR 2068967, DOI https://doi.org/10.1215/S0012-7094-04-12333-4
- Hans Grauert and Oswald Riemenschneider, VerschwindungssĂ€tze fĂŒr analytische Kohomologiegruppen auf komplexen RĂ€umen, Invent. Math. 11 (1970), 263â292 (German). MR 302938, DOI https://doi.org/10.1007/BF01403182
- Masaki Kashiwara, $B$-functions and holonomic systems. Rationality of roots of $B$-functions, Invent. Math. 38 (1976/77), no. 1, 33â53. MR 430304, DOI https://doi.org/10.1007/BF01390168
- M. Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 134â142. MR 726425, DOI https://doi.org/10.1007/BFb0099962
- Masaki Kashiwara and Takahiro Kawai, Second-microlocalization and asymptotic expansions, Complex analysis, microlocal calculus and relativistic quantum theory (Proc. Internat. Colloq., Centre Phys., Les Houches, 1979) Lecture Notes in Phys., vol. 126, Springer, Berlin-New York, 1980, pp. 21â76. MR 579740
- JĂĄnos KollĂĄr, Singularities of pairs, Algebraic geometryâSanta Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221â287. MR 1492525
- François Loeser, Quelques consĂ©quences locales de la thĂ©orie de Hodge, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 1, 75â92 (French). MR 781779
- B. Malgrange, Le polynĂŽme de Bernstein dâune singularitĂ© isolĂ©e, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974) Springer, Berlin, 1975, pp. 98â119. Lecture Notes in Math., Vol. 459 (French). MR 0419827
- B. Malgrange, PolynĂŽmes de Bernstein-Sato et cohomologie Ă©vanescente, Analysis and topology on singular spaces, II, III (Luminy, 1981) AstĂ©risque, vol. 101, Soc. Math. France, Paris, 1983, pp. 243â267 (French). MR 737934
- Alan Michael Nadel, Multiplier ideal sheaves and KĂ€hler-Einstein metrics of positive scalar curvature, Ann. of Math. (2) 132 (1990), no. 3, 549â596. MR 1078269, DOI https://doi.org/10.2307/1971429
- V. Navarro Aznar, Sur la thĂ©orie de Hodge-Deligne, Invent. Math. 90 (1987), no. 1, 11â76 (French). MR 906579, DOI https://doi.org/10.1007/BF01389031
- Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849â995 (1989) (French). MR 1000123, DOI https://doi.org/10.2977/prims/1195173930
- Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221â333. MR 1047415, DOI https://doi.org/10.2977/prims/1195171082
- Morihiko Saito, On $b$-function, spectrum and rational singularity, Math. Ann. 295 (1993), no. 1, 51â74. MR 1198841, DOI https://doi.org/10.1007/BF01444876
- J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525â563. MR 0485870
- J. H. M. Steenbrink, The spectrum of hypersurface singularities, AstĂ©risque 179-180 (1989), 11, 163â184. Actes du Colloque de ThĂ©orie de Hodge (Luminy, 1987). MR 1042806
28 VaquiĂ©, M., IrrĂ©gularitĂ© des revĂȘtements cycliques, in Singularities (Lille, 1991), London Math. Soc. Lecture Note Ser., 201, Cambridge Univ. Press, Cambridge, 1994, pp. 383â419.
29 Varchenko, A., Asymptotic Hodge structure in the vanishing cohomology, Math. USSR Izv. 18 (1982), 465â512.
1 Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux pervers, Astérisque 100, Soc. Math. France, Paris, 1982.
2 Borel, A. et al., Algebraic ${\mathcal {D}}$-modules, Perspectives in Math. 2, Academic Press, 1987.
3 Budur, N., thesis.
4 ---, On Hodge spectrum and multiplier ideals, Math. Ann. 327 (2003), no. 2, 257â270.
5 ---, Multiplier ideals and filtered $D$-modules, preprint.
6 Deligne, P., ThĂ©orie de Hodge I, Actes CongrĂšs Intern. Math., Part 1 (1970), 425â430; II, Publ. Math. IHES, 40 (1971), 5â58; III, ibid. 44 (1974), 5â77.
7 ---, Le formalisme des cycles Ă©vanescents, in SGA7 XIII and XIV, Lect. Notes in Math. 340, Springer, Berlin, 1973, pp. 82â115 and 116â164.
8 Demailly, J.-P., $L^{2}$-Methods and effective results in algebraic geometry, Proc. Intern. Congr. Math. (ZĂŒrich, 1994), BirkhĂ€user, Basel, 1995, pp. 817â827.
9 Denef, J. and Loeser, F., Motivic Igusa zeta functions, J. Alg. Geom. 7 (1998), 505â537.
10 Ein, L., Multiplier ideals, vanishing theorems and applications, Proc. Symp. Pure Math., A.M.S. 62 Part 1, (1997), 203â219.
11 Ein, L. and Lazarsfeld, R., Singularities of theta divisors and the birational geometry of irregular varieties, J. Amer. Math. Soc. 10 (1997), 243â258.
12 Ein, L., Lazarsfeld, R., Smith, K.E. and Varolin, D., Jumping coefficients of multiplier ideals, Duke Math. J. 123 (2004), no. 3, 469â506.
13 Grauert, H. and Riemenschneider, O., VerschwindungssĂ€tze fĂŒr analytische Kohomologiegruppen auf Komplexen RĂ€umen, Inv. Math. 11 (1970), 263â292.
14 Kashiwara, M., $B$-functions and holonomic systems, Inv. Math. 38 (1976/77), 33â53.
15 ---, Vanishing cycle sheaves and holonomic systems of differential equations, Algebraic geometry (Tokyo/Kyoto, 1982), Lect. Notes in Math. 1016, Springer, Berlin, 1983, pp. 134â142.
16 Kashiwara, M. and Kawai, T., Second-microlocalization and asymptotic expansions, Proc. Internat. Colloq., Centre Phys. (Les Houches, 1979), Lecture Notes in Phys., 126, Springer, Berlin, 1980, pp. 21â76.
17 KollĂĄr, J., Singularities of pairs, Proc. Symp. Pure Math., A.M.S. 62 Part 1, (1997), 221â287.
18 Loeser, F., Quelques consĂ©quences locales de la thĂ©orie de Hodge, Ann. Inst. Fourier 35 (1985) 75â92.
19 Malgrange, B., Le polynĂŽme de Bernstein dâune singularitĂ© isolĂ©e, in Lect. Notes in Math. 459, Springer, Berlin, 1975, pp. 98â119.
20 ---, PolynĂŽme de Bernstein-Sato et cohomologie Ă©vanescente, Analysis and topology on singular spaces, II, III (Luminy, 1981), AstĂ©risque 101â102 (1983), 243â267.
21 Nadel, A.M., Multiplier ideal sheaves and KĂ€hler-Einstein metrics of positive scalar curvature, Ann. Math. 132 (1990), 549â596.
22 Navarro Aznar, V., Sur la thĂ©orie de Hodge-Deligne, Inv. Math. 90 (1987), 11â76.
23 Saito, M., Modules de Hodge polarisables, Publ. RIMS, Kyoto Univ. 24 (1988), 849â995.
24 ---, Mixed Hodge modules, Publ. RIMS, Kyoto Univ. 26 (1990), 221â333.
25 ---, On $b$-function, spectrum and rational singularity, Math. Ann. 295 (1993), 51â74.
26 Steenbrink, J.H.M., Mixed Hodge structure on the vanishing cohomology, in Real and Complex Singularities (Proc. Nordic Summer School, Oslo, 1976) Alphen a/d Rijn: Sijthoff & Noordhoff 1977, pp. 525â563.
27 ---, The spectrum of hypersurface singularity, AstĂ©risque 179â180 (1989), 163â184.
28 VaquiĂ©, M., IrrĂ©gularitĂ© des revĂȘtements cycliques, in Singularities (Lille, 1991), London Math. Soc. Lecture Note Ser., 201, Cambridge Univ. Press, Cambridge, 1994, pp. 383â419.
29 Varchenko, A., Asymptotic Hodge structure in the vanishing cohomology, Math. USSR Izv. 18 (1982), 465â512.
Additional Information
Nero Budur
Affiliation:
Department of Mathematics, University of Illinois at Chicago, 851 South Morgan Street (M/C 249), Chicago, Illinois 60607-7045
Address at time of publication:
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email:
nero@math.uic.edu, nbudur@math.jhu.edu
Morihiko Saito
Affiliation:
RIMS Kyoto University, Kyoto 606-8502 Japan
Email:
msaito@kurims.kyoto-u.ac.jp
Received by editor(s):
June 15, 2003
Published electronically:
December 9, 2004