Valuations and multiplier ideals

Favre, Charles; Jonsson, Mattias

doi:10.1090/S0894-0347-05-00481-9

Valuations and multiplier ideals

Authors: Charles Favre and Mattias Jonsson
Journal: J. Amer. Math. Soc. 18 (2005), 655-684
MSC (2000): Primary 14B05; Secondary 32U25, 13H05
DOI: https://doi.org/10.1090/S0894-0347-05-00481-9
Published electronically: April 13, 2005
MathSciNet review: 2138140
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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.

[BM] M. Blel and S.K. Mimouni.
Singularités et intégrabilité des fonctions plurisousharmoniques.
Ann. Inst. Fourier, to appear.
[D] Jean-Pierre Demailly, Multiplier ideal sheaves and analytic methods in algebraic geometry, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) ICTP Lect. Notes, vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp. 1–148. MR 1919457
[DEL] Jean-Pierre Demailly, Lawrence Ein, and Robert Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137–156. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786484, https://doi.org/10.1307/mmj/1030132712
[DK] Jean-Pierre Demailly and János 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 (English, with English and French summaries). MR 1852009, https://doi.org/10.1016/S0012-9593(01)01069-2
[ELS] Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003), no. 2, 409–440. MR 1963690
[FJ1] Charles Favre and Mattias Jonsson, The valuative tree, Lecture Notes in Mathematics, vol. 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, Iwanami Shoten, Tokyo, 1977, pp. 357–368. MR 0485881
[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] Christer O. Kiselman, Attenuating the singularities of plurisubharmonic functions, Ann. Polon. Math. 60 (1994), no. 2, 173–197 (English, with English and Esperanto summaries). MR 1301603, https://doi.org/10.4064/ap-60-2-173-197
[Ku] Takayasu Kuwata, On log canonical thresholds of reducible plane curves, Amer. J. Math. 121 (1999), no. 4, 701–721. MR 1704476
[La] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
[Li] Joseph Lipman, Adjoints of ideals in regular local rings, Math. Res. Lett. 1 (1994), no. 6, 739–755. With an appendix by Steven Dale Cutkosky. MR 1306018, https://doi.org/10.4310/MRL.1994.v1.n6.a10
[LW] Joseph Lipman and Kei-ichi Watanabe, Integrally closed ideals in two-dimensional regular local rings are multiplier ideals, Math. Res. Lett. 10 (2003), no. 4, 423–434. MR 1995782, https://doi.org/10.4310/MRL.2003.v10.n4.a2
[MP] James McKernan and Yuri Prokhorov, Threefold thresholds, Manuscripta Math. 114 (2004), no. 3, 281–304. MR 2075967, https://doi.org/10.1007/s00229-004-0457-x
[Mi] S. K. Mimouni.
Singularités des fonctions plurisousharmoniques et courants de Liouville.
Thèse de la faculté des sciences de Monastir, Tunisie, 2001.
[N] Alan Michael Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Ann. of Math. (2) 132 (1990), no. 3, 549–596. MR 1078269, https://doi.org/10.2307/1971429
[PS] D. H. Phong and Jacob Sturm, On a conjecture of Demailly and Kollár, Asian J. Math. 4 (2000), no. 1, 221–226. Kodaira’s issue. MR 1803721, https://doi.org/10.4310/AJM.2000.v4.n1.a14
[S] V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203 (Russian); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95–202. MR 1162635, https://doi.org/10.1070/IM1993v040n01ABEH001862
[Sp] Mark Spivakovsky, Valuations in function fields of surfaces, Amer. J. Math. 112 (1990), no. 1, 107–156. MR 1037606, https://doi.org/10.2307/2374856
[S] Henri Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans 𝐶ⁿ, Bull. Soc. Math. France 100 (1972), 353–408 (French). MR 352517
[TW] Shunsuke Takagi and Kei-ichi Watanabe, When does the subadditivity theorem for multiplier ideals hold?, Trans. Amer. Math. Soc. 356 (2004), no. 10, 3951–3961. MR 2058513, https://doi.org/10.1090/S0002-9947-04-03436-1