|
Valuations and multiplier ideals
Author(s):
Charles
Favre;
Mattias
Jonsson
Journal:
J. Amer. Math. Soc.
18
(2005),
655-684.
MSC (2000):
Primary 14B05;
Secondary 32U25, 13H05
Posted:
April 13, 2005
MathSciNet review:
2138140
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are a formula for the complex integrability exponent of a plurisubharmonic function in terms of Kiselman numbers, and a proof of the openness conjecture by Demailly and Kollár. Our technique also yields new proofs of two recent results: one on the structure of the set of complex singularity exponents for holomorphic functions; the other by Lipman and Watanabe on the realization of ideals as multiplier ideals.
References:
-
- [BM]
- M. Blel and S.K. Mimouni.
Singularités et intégrabilité des fonctions plurisousharmoniques.
Ann. Inst. Fourier, to appear. - [D]
- J.-P. Demailly.
Multiplier ideal sheaves and analytic methods in algebraic geometry.
In School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste 2000), 1-148.
ICTP Lect. Notes, vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. MR 1919457 (2003f:32020) - [DEL]
- J.-P. Demailly, L. Ein and R. Lazarsfeld.
A subadditivity property of multiplier ideals.
Michigan Math. J. 48 (2000), 137-156. MR 1786484 (2002a:14016) - [DK]
- J.-P. Demailly and J. Kollár.
Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds.
Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 4, 525-556. MR 1852009 (2002e:32032) - [ELS]
- L. Ein, R. Lazarsfeld and K.E. Smith.
Uniform approximation of Abhyankar valuation ideals in smooth function fields.
Amer. J. Math. 125 (2003), no. 2, 409-440. MR 1963690 (2003m:13004) - [FJ1]
- C. Favre and M. Jonsson.
The valuative tree.
Lecture Notes in Mathematics, 1853. Springer-Verlag, Berlin, 2004. MR 2097722 - [FJ2]
- C. Favre and M. Jonsson.
Valuative analysis of planar plurisubharmonic functions.
Invent. Math., to appear. - [I]
- J. Igusa.
On the first terms of certain asymptotic expansions.
Complex analysis and algebraic geometry, 357-368.
Iwanami Shoten, Tokyo, 1977. MR 0485881 (58:5680) - [Ki1]
- C.-O. Kiselman.
Un nombre de Lelong raffiné.
Séminaire d'analyse complexe et géométrie 1985-1987, pp. 61-70. Faculté des sciences de Tunis et Faculté des Sciences et techniques de Monastir. - [Ki2]
- C.-O. Kiselman.
Attenuating the singularities of plurisubharmonic functions.
Ann. Polon. Math. 60 (1994), no. 2, 173-197. MR 1301603 (95i:32024) - [Ku]
- T. Kuwata.
On log canonical thresholds of reducible plane curves.
Amer. J. Math. 121 (1999), no. 4, 701-721. MR 1704476 (2001g:14047) - [La]
- R. Lazarsfeld.
Positivity in algebraic geometry II.
Ergebnisse der Math. und ihrer Grenzgebiete (3), vol. 49. Springer-Verlag, Berlin, 2004. MR 2095472 - [Li]
- J. Lipman.
Adjoints of ideals in regular local rings.
Math. Res. Lett. 1 (1994), no. 6, 739-755.
With an appendix by S. C. Cutkosky. MR 1306018 (95k:13028) - [LW]
- J. Lipman and K.-I. Watanabe.
Integrally closed ideals in two-dimensional regular local rings are multiplier ideals.
Math. Res. Lett. 10 (2003), no. 4, 423-434. MR 1995782 (2004m:13059) - [MP]
- J. McKernan and Y. Prokhorov.
Threefold thresholds.
Manuscripta Math. 114 (2004), 281-304. MR 2075967 - [Mi]
- S. K. Mimouni.
Singularités des fonctions plurisousharmoniques et courants de Liouville.
Thèse de la faculté des sciences de Monastir, Tunisie, 2001. - [N]
- A. M. Nadel.
Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature.
Ann. of Math. (2) 132 (1990), 549-596. MR 1078269 (92d:32038) - [PS]
- D.-H. Phong and J. Sturm.
On a conjecture of Demailly and Kollár.
Kodaira's issue. Asian J. Math. 4 (2000), no. 1, 221-226. MR 1803721 (2002b:32043) - [S]
- V. V. Shokurov.
Three-dimensional log perestroikas.
Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95-202. MR 1162635 (93j:14012) - [Sp]
- M. Spivakovsky.
Valuations in function fields of surfaces.
Amer. J. Math. 112 (1990), no. 1, 107-156. MR 1037606 (91c:14037) - [S]
- H. Skoda.
Sous-ensembles analytiques d'ordre fini ou infini dans  .
Bull. Soc. Math. France 100 (1972), 353-408. MR 0352517 (50:5004) - [TW]
- S. Takagi and K.-I. Watanabe.
When does the subadditivity theorem for multiplier ideals hold?
Trans. Amer. Math. Soc. 356 (2004), 3951-3961. MR 2058513 (2005d:13016)
Similar Articles:
Retrieve articles in Journal of the American Mathematical Society
with
MSC (2000):
14B05,
32U25, 13H05
Retrieve articles in all Journals with
MSC (2000):
14B05,
32U25, 13H05
Additional Information:
Charles
Favre
Affiliation:
CNRS, Institut de Mathématiques, Equipe Géométrie et Dynamique, F-75251 Paris Cedex 05, France
Email:
favre@math.jussieu.fr
Mattias
Jonsson
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
Address at time of publication:
Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Email:
mattiasj@umich.edu, mattiasj@kth.se
DOI:
10.1090/S0894-0347-05-00481-9
PII:
S 0894-0347(05)00481-9
Keywords:
Valuations,
multiplier ideals,
singularity exponents,
Arnold multiplicity,
Lelong numbers,
Kiselman numbers,
trees,
Laplace operator.
Received by editor(s):
January 16, 2004
Posted:
April 13, 2005
Additional Notes:
The second author was partially supported by NSF Grant No. DMS-0200614
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
