Algebraic bounds on analytic multiplier ideals

Lehmann, Brian

Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 1077-1108.

Pour un diviseur pseudo-effectif $L$ nous construisons l’idéal diminué ${𝒥}_{\sigma }\left(L\right)$ 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 ${𝒪}_X\left(L\right)$ est souvent contenu dans ${𝒥}_{\sigma }\left(L\right)$. Nous caractérisons les diviseurs abondants par l’idéal diminué, montrant que les informations de nature géométriques et analytique doivent coïncider.

Given a pseudo-effective divisor $L$ we construct the diminished ideal ${𝒥}_{\sigma }\left(L\right)$, 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 $𝒥\left(h_{\min }\right)$ of the metric of minimal singularities on ${𝒪}_X\left(L\right)$ is contained in ${𝒥}_{\sigma }\left(L\right)$. We also characterize abundant divisors using the diminished ideal, indicating that the geometric and analytic information should coincide.

Classification : 14C20
Keywords: Multiplier ideals, metric of minimal singularities
Mot clés : idéal multiplicateur, métrique à singularités minimales

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

[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., Volume 22 (2013), pp. 201-248 | MR | Zbl

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

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

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

[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., Volume 48 (2000), pp. 137-156 | MR | Zbl

[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), Volume 34 (2001) no. 4, pp. 525-556 | Numdam | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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, MSJ Memoirs, 14, Mathematical Society of Japan, Tokyo, 2004 | MR | Zbl

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

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

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