Algebraic bounds on analytic multiplier ideals
Annales de l'Institut Fourier, Volume 64 (2014) no. 3, p. 1077-1108

Given a pseudo-effective divisor L we construct the diminished ideal 𝒥 σ (L), a “continuous” extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. Our main theorem shows that for most pseudo-effective divisors L the multiplier ideal 𝒥(h min ) of the metric of minimal singularities on 𝒪 X (L) is contained in 𝒥 σ (L). We also characterize abundant divisors using the diminished ideal, indicating that the geometric and analytic information should coincide.

Pour un diviseur pseudo-effectif L nous construisons l’idéal diminué 𝒥 σ (L) qui est une extension “continue” de l’idéal multiplicateur asymptotique pour les grands diviseurs au cône pseudo-effectif. L’idéal multiplicateur d’une métrique hermitiennes à singularités minimales sur 𝒪 X (L) est souvent contenu dans 𝒥 σ (L). Nous caractérisons les diviseurs abondants par l’idéal diminué, montrant que les informations de nature géométriques et analytique doivent coïncider.

DOI : https://doi.org/10.5802/aif.2874
Classification:  14C20
Keywords: Multiplier ideals, metric of minimal singularities
@article{AIF_2014__64_3_1077_0,
     author = {Lehmann, Brian},
     title = {Algebraic bounds on analytic multiplier ideals},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {3},
     year = {2014},
     pages = {1077-1108},
     doi = {10.5802/aif.2874},
     mrnumber = {3330164},
     zbl = {06387301},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2014__64_3_1077_0}
}
Lehmann, Brian. Algebraic bounds on analytic multiplier ideals. Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1077-1108. doi : 10.5802/aif.2874. http://www.numdam.org/item/AIF_2014__64_3_1077_0/

[1] Boucksom, S. Divisorial Zariski Decompositions on Compact Complex Manifolds, Ann. Sci. École Norm. Sup., Tome 37 (2004) no. 4, pp. 45-76 | Numdam | MR 2050205 | Zbl 1054.32010

[2] Boucksom, S.; Demailly, J.-P.; Păun, M.; Peternell, T. The Pseudo-Effective Cone of a Compact Kähler Manifold and Varieties of Negative Kodaira Dimension, J. Alg. Geom., Tome 22 (2013), pp. 201-248 | MR 3019449 | Zbl 1267.32017

[3] Boucksom, S.; Favre, C.; Jonsson, M. Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci., Tome 44 (2008) no. 2, pp. 449-494 | MR 2426355 | Zbl 1146.32017

[4] Demailly, J.-P. Singular Hermitian metrics on positive line bundles, Complex Algebraic Varieties (Bayreuth, 1990), Springer, Berlin (Lecture Notes in Math., 1507) (1992), pp. 87-104 | MR 1178721 | Zbl 0784.32024

[5] Demailly, J.-P. Monge-Ampère Operators, Lelong Numbers and Intersection Theory, Complex Analysis and Geometry, Plenum Press, New York (Univ. Ser. Math.) (1993), pp. 115-193 | MR 1211880 | Zbl 0792.32006

[6] Demailly, J.-P. Complex analytic and differential geometry (2012) (http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)

[7] Demailly, J.-P.; Ein, L.; Lazarsfeld, R. A Subadditivity Property of Multiplier Ideals, Michigan Math. J., Tome 48 (2000), pp. 137-156 | MR 1786484 | Zbl 1077.14516

[8] Demailly, J.-P.; Kollár, J. Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4), Tome 34 (2001) no. 4, pp. 525-556 | Numdam | MR 1852009 | Zbl 0994.32021

[9] Demailly, J.-P.; Peternell, T.; Schneider, M. Compact complex manifolds with numerically effective tangent bundles, Journal of Algebraic Geometry, Tome 3 (1994) no. 2, pp. 295-346 | MR 1257325 | Zbl 0827.14027

[10] Demailly, J.-P.; Peternell, T.; Schneider, M. Pseudo-effective line bundles on compact Kähler manifolds, Internat. J. Math., Tome 12 (2001) no. 6, pp. 689-741 | MR 1875649 | Zbl 1111.32302

[11] Eckl, T. Numerically trival foliations, Ann. Inst. Fourier (Grenoble), Tome 54 (2004) no. 4, pp. 887-938 | Numdam | MR 2111016 | Zbl 1071.32018

[12] Eckl, T. Numerically trival foliations, Iitaka fibrations, and the numerical dimension (2005) (arXiv:math/0508340v1) | Numdam | MR 2111016

[13] Ein, L.; Lazarsfeld, R.; Mustaţă, M.; Nakamaye, M.; Popa, M. Asymptotic invariants of base loci, Pure Appl. Math. Q., Tome 1 (2005) no. 2, pp. 379-403 | MR 2194730 | Zbl 1139.14008

[14] Ein, L.; Lazarsfeld, R.; Mustaţă, M.; Nakamaye, M.; Popa, M. Restricted Volumes and Base Loci of Linear Series, Amer. J. Math., Tome 131 (2009) no. 3, pp. 607-651 | MR 2530849 | Zbl 1179.14006

[15] Favre, C.; Jonsson, M. The valuative tree, Springer-Verlag, Berlin Heidelberg, Lecture notes in mathematics, Tome 1853 (2004) | MR 2097722 | Zbl 1064.14024

[16] Favre, C.; Jonsson, M. Valuations and multiplier ideals, J. Amer. Math. Soc., Tome 18 (2005) no. 3, pp. 655-684 | MR 2138140 | Zbl 1075.14001

[17] Hacon, C. A derived category approach to generic vanishing, J. Reine Angew. Math., Tome 575 (2004), pp. 173-187 | MR 2097552 | Zbl 1137.14012

[18] Jonsson, M.; Mustaţă, M. Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble), Tome 62 (2012) no. 6, pp. 2145-2209 | Numdam | MR 3060755 | Zbl 1272.14016

[19] Lazarsfeld, R. Positivity in Algebraic Geometry I-II, Springer-Verlag, Berlin Heidelberg, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Tome 48-49 (2004) | MR 2095471 | Zbl 0633.14016

[20] Lehmann, B. On Eckl’s pseudo-effective reduction map (2011) (arXiv:1103.1073, to appear in Trans. of the A.M.S.)

[21] Nakayama, N. Zariski-decomposition and abundance, Mathematical Society of Japan, Tokyo, MSJ Memoirs, Tome 14 (2004) | MR 2104208 | Zbl 1061.14018

[22] Russo, F. A characterization of nef and good divisors by asymptotic multiplier ideals, Bulletin of the Belgian Mathematical Society-Simon Stevin, Tome 16 (2009) no. 5, pp. 943-951 | MR 2574371 | Zbl 1183.14011

[23] Siu, Y.T. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Inventiones Mathematicae, Tome 27 (1974) no. 1, pp. 53-156 | MR 352516 | Zbl 0289.32003

[24] Takayama, S. Iitaka’s fibration via multiplier ideals, Trans. Amer. Math. Soc., Tome 355 (2003) no. 1, pp. 37-47 | MR 1928076 | Zbl 1055.14011