Abstract
In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function \({\varphi}\) with an isolated singularity at 0 in an open subset of \({\mathbb{C}^n}\). This threshold is defined as the supremum of constants c > 0 such that \({e^{-2c\varphi}}\) is integrable on a neighborhood of 0. We relate \({c(\varphi)}\) to the intermediate multiplicity numbers \({e_j(\varphi)}\), defined as the Lelong numbers of \({(dd^c\varphi)^j}\) at 0 (so that in particular \({e_0(\varphi)=1}\)). Our main result is that \({c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}\). This inequality is shown to be sharp; it simultaneously improves the classical result \({c(\varphi)\geqslant 1/e_1(\varphi)}\) due to Skoda, as well as the lower estimate \({c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}\) which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.
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References
Bedford E., Taylor B. A.: The Dirichlet problem for a complex Monge–Ampère equation. Invent. Math. 37, 1–44 (1976)
Bedford E, Taylor B.A: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)
Cegrell U.: The general definition of the complex Monge–Ampère operator. Ann. Inst. Fourier (Grenoble) 54, 159–179 (2004)
Chel′tsov, I. A., Birationally rigid Fano varieties. Uspekhi Mat. Nauk, 60:5 (2005), 71–160 (Russian); English translation in Russian Math. Surveys, 60 (2005), 875– 965.
Corti A.: Factoring birational maps of threefolds after Sarkisov. J. Algebraic Geom. 4, 223–254 (1995)
Corti, A., Singularities of linear systems and 3-fold birational geometry, in Explicit Birational Geometry of 3-folds, London Math. Soc. Lecture Note Ser., 281, pp. 259–312. Cambridge Univ. Press, Cambridge, 2000.
Demailly J.-P.: Nombres de Lelong généralisés, théorèmes d’intégralité et d’analyticité. Acta Math. 159, 153–169 (1987)
Demailly J.-P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361–409 (1992)
Demailly, J.-P., Monge–Ampère operators, Lelong numbers and intersection theory, in Complex Analysis and Geometry, Univ. Ser. Math., pp. 115–193. Plenum, New York, 1993.
Demailly, J.-P., Estimates on Monge–Ampère operators derived from a local algebra inequality, in Complex Analysis and Digital Geometry, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 86, pp. 131–143. Uppsala Universitet, Uppsala, 2009.
Demailly J.-P., Kollár J.: Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup, 34, 525–556 (2001)
Eisenbud, D., Commutative Algebra. Graduate Texts in Mathematics, 150. Springer, New York, 1995.
de Fernex T., Ein L., Mustaţă M.: Bounds for log canonical thresholds with applications to birational rigidity. Math. Res. Lett, 10, 219–236 (2003)
de Fernex T., Ein L., Mustaţă M.: Multiplicities and log canonical threshold. J. Algebraic Geom. 13, 603–615 (2004)
Howald J.A.: Multiplier ideals of monomial ideals. Trans. Amer. Math. Soc, 353, 2665–2671 (2001)
Iskovskikh, V.A., Birational rigidity of Fano hypersurfaces in the framework of Mori theory. Uspekhi Mat. Nauk, 56:2 (2001), 3–86 (Russian); English translation in Russian Math. Surveys, 56 (2001), 207–291.
Iskovskikh, V. A. & Manin, J. I., Three-dimensional quartics and counterexamples to the L¨uroth problem. Mat. Sb., 86(128) (1971), 140–166 (Russian); English translation in Math. USSR–Sb., 15 (1971), 141–166.
Kiselman, C.O., Un nombre de Lelong raffiné, in Séminaire d’Analyse Complexe et Géométrie 1985–87, pp. 61–70. Faculté des Sciences de Tunis & Faculté des Sciences et Techniques de Monastir, Monastir, 1987.
Kiselman C.O.: Attenuating the singularities of plurisubharmonic functions. Ann. Polon. Math, 60, 173–197 (1994)
Ohsawa T., Takegoshi K.: On the extension of L 2 holomorphic functions. Math. Z, 195, 197–204 (1987)
Pukhlikov A. V.: Birational isomorphisms of four-dimensional quintics. Invent. Math. 87, 303–329 (1987)
Pukhlikov, A. V., Birationally rigid Fano hypersurfaces. Izυ. Ross. Akad. Nauk Ser. Mat., 66:6 (2002), 159–186 (Russian); English translation in Izυ. Math., 66 (2002), 1243–1269.
Skoda H.: Sous-ensembles analytiques d’ordre fini ou infini dans C n. Bull. Soc. Math. France, 100, 353–408 (1972)
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Demailly, JP., Phạm, H.H. A sharp lower bound for the log canonical threshold. Acta Math 212, 1–9 (2014). https://doi.org/10.1007/s11511-014-0107-4
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DOI: https://doi.org/10.1007/s11511-014-0107-4