Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-08T14:26:49.326Z Has data issue: false hasContentIssue false
  • English
  • Français

An algebraic approach to the openness conjecture of Demailly and Kollár

Part of: Pluripotential theory

Published online by Cambridge University Press:  11 March 2013

Mattias Jonsson  and
Mircea Mustaţă
Mattias Jonsson
Affiliation:
Dept of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA ([email protected]; [email protected])
Mircea Mustaţă
Affiliation:
Dept of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We reduce the openness conjecture of Demailly and Kollár on the singularities of plurisubharmonic functions to a purely algebraic statement.

Keywords

plurisubharmonic functiongraded sequencelog canonical thresholdvaluation

MSC classification

Secondary: 32U05: Plurisubharmonic functions and generalizations 32U25: Lelong numbers
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 1 , January 2014 , pp. 119 - 144
Copyright
©Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berndtsson, B., Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains, Ann. Inst. Fourier 56 (2006), 1633–1662.Google Scholar
Boucksom, S., Favre, C. and Jonsson, M., Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci. 44 (2008), 449–494.Google Scholar
Demailly, J.-P., Complex analytic and algebraic geometry (book available at www-fourier.ujf-grenoble.fr/~demailly).Google Scholar
Demailly, J.-P., Nombres de Lelong généralisés, théorèmes d’intégralité et d’analyticité, Acta Math. 157 (1987), 153–169.Google Scholar
Demailly, J.-P., Regularization of closed positive currents and intersection theory, J. Algebra Geom. 1 (1992), 361–409.Google Scholar
Demailly, J.-P., A numerical criterion for very ample line bundles, J. Differential Geom. 37 (1993), 323–374.Google Scholar
Demailly, J.-P., Ein, L. and Lazarsfeld, R., A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137–156.Google Scholar
Demailly, J.-P. and Kollár, J., Semicontinuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), 525–556.Google Scholar
Ein, L., Lazarsfeld, R. and Smith, K. E., Uniform approximation of Abhyankar valuations in smooth function fields, Amer. J. Math. 125 (2003), 409–440.Google Scholar
Ein, L., Lazarsfeld, R., Smith, K. E. and Varolin, D., Jumping coefficients of multiplier ideals, Duke Math. J. 123 (2004), 469–506.Google Scholar
Favre, C. and Jonsson, M., The Valuative Tree, Lecture Notes in Mathematics, Volume 1853 (Springer-Verlag, Berlin, 2004).Google Scholar
Favre, C. and Jonsson, M., Valuative analysis of planar plurisubharmonic functions, Invent. Math. 162 (2005), 271–311.Google Scholar
Favre, C. and Jonsson, M., Valuations and multiplier ideals, J. Amer. Math. Soc. 18 (2005), 655–684.CrossRefGoogle Scholar
Guenancia, H., Toric plurisubharmonic functions and analytic adjoint ideal sheaves (arXiv:1011.3162).Google Scholar
Hörmander, L., Notions of Convexity, Progress in Mathematics, Volume 127 (Birkhäuser, Boston, MA, 1994).Google Scholar
Hu, Z., Valuative multiplier ideals, preprint.Google Scholar
Hu, Z., Valuations and log canonical thresholds, preprint.Google Scholar
Jonsson, M., Dynamics on Berkovich spaces in low dimensions, in Berkovich spaces and applications, Séminaires et Congrès, Société Mathématique de France, in press (arXiv:1201.1944).Google Scholar
Jonsson, M. and Mustaţă, M., Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier 62 (2012), 2145–2209.Google Scholar
Kiselman, C.-O., Un nombre de Lelong raffiné, in Séminaire d’analyse complexe et géométrie 1985–1987, Faculté des sciences de Tunis et Faculté des Sciences et Techniques de Monastir, 1987, pp. 61–70.Google Scholar
Kiselman, C. O., Attenuating the singularities of plurisubharmonic functions, Ann. Polon. Math. 60 (1994), 173–197.CrossRefGoogle Scholar
Lagerberg, A., A new generalization of the Lelong number (arXiv:1001.3562).Google Scholar
Lazarsfeld, R., Positivity in Algebraic Geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Volume 49 (Springer-Verlag, Berlin, 2004).Google Scholar
Lelong, P., Plurisubharmonic Functions and Positive Differential Forms (Gordon and Breach, New York, and Dunod, Paris, 1969).Google Scholar
Matsumura, H., Commutative Algebra, Mathematics Lecture Note Series, Volume 56 (Benjamin/Cummings, Reading, Mass, 1980).Google Scholar
Mustaţă, M., On multiplicities of graded sequences of ideals, J. Algebra 256 (2002), 229–249.CrossRefGoogle Scholar
Nadel, A. M., Multiplier ideal sheaves and existence of Kähler–Einstein metrics of positive scalar curvature, Proc. Natl Acad. Sci. USA 86 (1989), 7299–7300.CrossRefGoogle ScholarPubMed
Nadel, A. M., Multiplier ideal sheaves and Kähler–Einstein metrics of positive scalar curvature, Ann. of Math. 132 (1990), 549–596.Google Scholar
Rashkovskii, A., Relative types and extremal problems for plurisubharmonic functions. Int. Math. Res. Not. 2006, Art. ID 76283.Google Scholar
Siu, Y.-T., Analyticity of sets associated to Lelong numbers and the extension of positive closed currents, Invent. Math. 27 (1974), 53–156.Google Scholar
Skoda, H., Sous-ensembles analytiques d’ordre fini ou infini dans ${\mathbf{C} }^{n} $, Bull. Soc. Math. France 100 (1972), 353–408.Google Scholar