Complex analysis
Extremal cases for the log canonical threshold
Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 21-24.

We show that a recent result of Demailly and Pham Hoang Hiep [12] implies a description of plurisubharmonic functions with given Monge–Ampère mass and smallest possible log canonical threshold. We also study an equality case for the inequality from [12].

Nous montrons qu'un résultat récent de Demailly et Pham Hoang Hiep [12] implique une description des fonctions plurisousharmoniques avec une masse de Monge–Ampère donnée et le seuil log-canonique le plus petit possible. Nous étudions aussi le cas d'égalité dans l'inégalité de [12].

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DOI: 10.1016/j.crma.2014.11.002
Rashkovskii, Alexander 1

1 Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway
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Rashkovskii, Alexander. Extremal cases for the log canonical threshold. Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 21-24. doi : 10.1016/j.crma.2014.11.002. http://www.numdam.org/articles/10.1016/j.crma.2014.11.002/

[1] Åhag, P.; Cegrell, U.; Czyz, R.; Pham Hoàng Hiep Monge–Ampère measures on pluripolar sets, J. Math. Pures Appl. (9), Volume 92 (2009) no. 6, pp. 613-627

[2] Åhag, P.; Cegrell, U.; Kolodziej, S.; Pham, H.H.; Zeriahi, A. Partial pluricomplex energy and integrability exponents of plurisubharmonic functions, Adv. Math., Volume 222 (2009) no. 6, pp. 2036-2058

[3] Berndtsson, B. Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 6, pp. 1633-1662

[4] Bivià-Ausina, C. Joint reductions of monomial ideals and multiplicity of complex analytic maps, Math. Res. Lett., Volume 15 (2008) no. 2, pp. 389-407

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

[6] Cegrell, U. The general definition of the complex Monge–Ampère operator, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 1, pp. 159-179

[7] Cegrell, U. Convergence in capacity, Can. Math. Bull., Volume 55 (2012) no. 2, pp. 242-248

[8] Cegrell, U.; Zeriahi, A. Subextension of plurisubharmonic functions with bounded Monge–Ampère mass, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003) no. 4, pp. 305-308

[9] de Fernex, T.; Ein, L.; Mustaţǎ, M. Multiplicities and log canonical threshold, J. Algeb. Geom., Volume 13 (2004) no. 3, pp. 603-615

[10] Demailly, J.-P. Monge–Ampère operators, Lelong numbers and intersection theory (Ancona, V.; Silva, A., eds.), Complex Analysis and Geometry, Univ. Ser. Math., Plenum Press, New York, 1993, pp. 115-193

[11] Demailly, J.-P. Estimates on Monge–Ampère operators derived from a local algebra inequality, 2006 (Passare, M., ed.), Uppsala University, Uppsala, Sweden (2009), pp. 131-143

[12] Demailly, J.-P.; Pham Hoàng Hiep A sharp lower bound for the log canonical threshold, Acta Math., Volume 212 (2014) no. 1, pp. 1-9

[13] Demailly, J.-P.; Kollár, J. Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Super. (4), Volume 34 (2001) no. 4, pp. 525-556

[14] Favre, C.; Jonsson, M. Valuative analysis of planar plurisubharmonic functions, Invent. Math., Volume 162 (2005), pp. 271-311

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

[16] Pham Hoàng Hiep A comparison principle for the log canonical threshold, C. R. Acad. Sci. Paris, Ser. I, Volume 351 (2013), pp. 441-443

[17] Kiselman, C.O. Attenuating the singularities of plurisubharmonic functions, Ann. Pol. Math., Volume LX.2 (1994), pp. 173-197

[18] Mustaţǎ, M. On multiplicities of graded sequences of ideals, J. Algebra, Volume 256 (2002) no. 1, pp. 229-249

[19] Rashkovskii, A. Newton numbers and residual measures of plurisubharmonic functions, Ann. Pol. Math., Volume 75 (2000) no. 3, pp. 213-231

[20] Rashkovskii, A. Relative types and extremal problems for plurisubharmonic functions, Int. Math. Res. Not. (2006) (26 p., Art. ID 76283)

[21] Rashkovskii, A. Multi-circled singularities, Lelong numbers, and integrability index, J. Geom. Anal., Volume 23 (2013) no. 4, pp. 1976-1992

[22] Skoda, H. Sous-ensembles analytiques d'ordre fini ou infini dans Cn, Bull. Soc. Math. Fr., Volume 100 (1972), pp. 353-408

[23] Zahariuta, V.P. Spaces of analytic functions and maximal plurisubharmonic functions, 1984 (D.Sci. dissertation, Rostov-on-Don, USSR)

[24] Zeriahi, A. Appendix: a stronger version of Demailly's estimate on Monge–Ampère operators, 2006 (Passare, M., ed.), Uppsala University, Uppsala, Sweden (2009), pp. 144-146

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