Valuations and asymptotic invariants for sequences of ideals

Jonsson, Mattias; Mustaţă, Mircea

Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2145-2209.

On étudie les nombres de saut asymptotiques pour les suites graduées d’idéaux et on démontre que ces invariants se calculent par une valuation réelle définie sur un corps de fonctions. Nous conjecturons que toute valuation qui calcule un tel nombre de saut est nécessairement quasi-monomiale. Cette conjecture est vraie en dimension deux. En général, on réduit la conjecture au cas de l’espace affine et des suites graduées d’idéaux de valuations. Au passage on étudie la structure d’un espace adéquat de valuations.

We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping number is necessarily quasi-monomial. This conjecture holds in dimension two. In general, we reduce it to the case of affine space and to graded sequences of valuation ideals. Along the way, we study the structure of a suitable valuation space.

[1] André, M. Localisation de la lissité formelle, Manuscripta Math., Volume 13 (1974), pp. 297-307 | MR | Zbl

[2] Baker, M.; Rumely, R. Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs, 159, American Mathematical Society, Providence, RI, 2010 | MR | Zbl

[3] Berkovich, V. G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, Amer. Math. Soc., Providence, RI, 1990 | MR | Zbl

[4] Berkovich, V. G. A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures, Algebra, Arithmetic and Geometry (Volume I: In Honor of Y. I. Manin, Progress in Mathematics), Volume 269, Birkhäuser Boston, 2010, pp. 49-67 | MR | Zbl

[5] Boucksom, S.; Favre, C.; Jonsson, M. Izumi’s theorem and non-Archimedean plurisubharmonic functions (In preparation)

[6] Boucksom, S.; Favre, C.; Jonsson, M. Pluripotential theory on valuation space (In preparation)

[7] Boucksom, S.; Favre, C.; Jonsson, M. Singular semipositive metrics in non-Archimedean geometry (arXiv:1201.0187)

[8] Boucksom, S.; Favre, C.; Jonsson, M. Solution to a non-Archimedean Monge-Ampère equation (arXiv:1201.0188)

[9] Boucksom, S.; Favre, C.; Jonsson, M. Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci., Volume 44 (2008), pp. 449-494 | MR | Zbl

[10] Boucksom, S.; de Fernex, T.; Favre, C. The volume of an isolated singularity (arXiv:1011.2847)

[11] Brøndsted, A. An introduction to convex polytopes, Graduate Texts in Mathematics, 90, Springer-Verlag, New York-Berlin, 1983 | MR | Zbl

[12] Conrad, B. Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc., Volume 22 (2007), pp. 205-257 | MR | Zbl

[13] Demailly, J.-P.; Ein, L.; Lazarsfeld, R. A subadditivity property of multiplier ideals, Michigan Math. J., Volume 48 (2000), pp. 137-156 | MR | Zbl

[14] Demailly, J.-P.; Kollár, J. Semicontinuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Supér. (4), Volume 34 (2001), pp. 525-556 | Numdam | MR | Zbl

[15] Ein, L.; Lazarsfeld, R.; Mustaţă, M.; Nakamaye, M.; Popa, M. Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble), Volume 56 (2006), pp. 1701-1734 | Numdam | MR | Zbl

[16] Ein, L.; Lazarsfeld, R.; Smith, K. E. Uniform approximation of Abhyankar valuations in smooth function fields, Amer. J. Math., Volume 125 (2003), pp. 409-440 | MR | Zbl

[17] Ein, L.; Lazarsfeld, R.; Smith, K. E.; Varolin, D. Jumping coefficients of multiplier ideals, Duke Math. J., Volume 123 (2004), pp. 469-506 | MR | Zbl

[18] Ein, L.; Mustaţă, M. Invariants of singularities of pairs, International Congress of Mathematicians, Volume II (2006), pp. 583-602 | MR | Zbl

[19] Favre, C.; Jonsson, M. The valuative tree, Lecture Notes in Mathematics, 1853, Springer, 2004 | MR | Zbl

[20] Favre, C.; Jonsson, M. Valuations and multiplier ideals, J. Amer. Math. Soc., Volume 18 (2005), pp. 655-684 | MR | Zbl

[21] Favre, C.; Jonsson, M. Valuative analysis of planar plurisubharmonic functions, Invent. Math., Volume 162 (2005) no. 2, pp. 271-311 | MR | Zbl

[22] de Fernex, T.; Ein, L.; Mustaţă, M. Log canonical thresholds on varieties with bounded singularities, Classification of algebraic varieties, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 221-257 | MR | Zbl

[23] de Fernex, T.; Mustaţă, M. Limits of log canonical thresholds, Ann. Sci. École Norm. Supér. (4), Volume 42 (2009), pp. 491-515 | Numdam | MR | Zbl

[24] Fulton, W. Introduction to toric varieties, Ann. of Math. Stud., 131, The William H. Rover Lectures in Geometry, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl

[25] Guenancia, H. Toric plurisubharmonic functions and analytic adjoint ideal sheaves (arXiv:1011.3162v2)

[26] Howald, J. Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc., Volume 353 (2001), pp. 2665-2671 | MR | Zbl

[27] Izumi, S. A measure of integrity for local analytic algebras, Publ. RIMS Kyoto Univ., Volume 21 (1985), pp. 719-735 | MR | Zbl

[28] Jonsson, M. Dynamics on Berkovich spaces in low dimensions (arXiv:1201.1944)

[29] Kedlaya, K. Good formal structures for flat meromorphic connections, I: Surfaces, Duke Math. J., Volume 154 (2010), pp. 343-418 | MR | Zbl

[30] Kedlaya, K. Good formal structures for flat meromorphic connections, II: Excellent schemes, J. Amer. Math. Soc., Volume 24 (2011), pp. 183-229 | MR

[31] Kempf, G.; Knudsen, F. F.; D., Mumford; Saint-Donat, B. Toroidal embeddings. I, Lecture Notes in Mathematics, 339, Springer-Verlag, Berlin, 1973 | MR | Zbl

[32] Kollár, J. Singularities of pairs, Algebraic geometry, Santa Cruz 1995 (Proc. Symp. Pure Math. 62, Part 1), Amer. Math. Soc., Providence, RI, 1997, pp. 221-286 | MR | Zbl

[33] Kontsevich, M.; Soibelman, Y. Affine structures and non-Archimedean analytic spaces, The unity of mathematics (Progr. Math.), Volume 244, Birkhäuser, Boston, 2006, pp. 321-385 | MR | Zbl

[34] Lazarsfeld, R. Positivity in algebraic geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 49, Springer-Verlag, Berlin, 2004 | MR | Zbl

[35] Matsumura, H. Commutative ring theory, translated from the Japanese by M. Reid, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1989 | MR | Zbl

[36] McNeal, J. D.; Zeytuncu, Y. E. Multiplier ideals and integral closure of monomial ideals: An analytic approach (arXiv:1001.4983) | Zbl

[37] Mustaţă, M. On multiplicities of graded sequences of ideals, J. Algebra, Volume 256 (2002), pp. 229-249 | MR | Zbl

[38] Payne, S. Analytification is the limit of all tropicalizations, Math. Res. Lett., Volume 16 (2009), pp. 543-556 | MR | Zbl

[39] Spivakovsky, M. Valuations in function fields of surfaces, Amer. J. Math., Volume 112 (1990), pp. 107-156 | MR | Zbl

[40] Temkin, M. Functorial desingularization over $\mathbf{Q}$: boundaries and the embedded case (arXiv:0912.2570)

[41] Temkin, M. Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math., Volume 219 (2008), pp. 488-522 | MR | Zbl

[42] Thuillier, A. Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov, University de Rennes 1 (2005) (Ph. D. Thesis tel.archives-ouvertes.fr/docs/00/04/87/50/PDF/tel-00010990.pdf)

[43] Thuillier, A. Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels, Manuscripta Math., Volume 123 (2007) no. 4, pp. 381-541 | MR | Zbl

[44] Tougeron, J.-C. Idéaux de fonctions differentiables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71, Springer-Verlag, Berlin, 1972 | MR | Zbl

[45] Vaquié, M. Valuations, Resolution of singularities (Obergurgl, 1997) (Progr. Math.), Volume 181, Birkhäuser, Basel, 2000, pp. 539-590 | MR | Zbl

[46] Wolfe, A. Cones and asymptotic invariants of multigraded systems of ideals, J. Algebra, Volume 319 (2008), pp. 1851-1869 | MR | Zbl

[47] Zariski, O. Local uniformization on algebraic varieties, Ann. of Math. (2), Volume 41 (1940), pp. 852-896 | MR | Zbl

[48] Zariski, O.; Samuel, P. Commutative algebra, II, Princeton, NJ, Van Nostrand, 1960 | MR | Zbl