Lelong numbers, complex singularity exponents, and Siu’s semicontinuity theorem. (Nombres de Lelong, exposants de singularités complexes et théorème de semi-continuité de Siu.) (English. French summary) Zbl 1370.32015
Let \(\varphi\) be a plurisubharmonic (psh) function and denote by \(\nu(\varphi,z_0)\) and \(c_{z_0}(\varphi)\), respectively, the Lelong number and the complex singularity exponent of \(\varphi\) at \(z_0\). The main result of the paper is the following: Let \(\varphi\) be a psh function on an open set \(\Omega\subset\mathbb C^n\). Then for any \(k\in\mathbb N\smallsetminus\{0\}\), there exists a psh function \(\varphi_k\) defined on a neighborhood of \(\Omega\times\{0\}\subset\mathbb C^n\times\mathbb C^k\) with coordinates \((z,w)\) such that
1. \(\varphi_k(z,0)=\varphi(z)\),
2. \(\nu(\varphi_k,(z,0))=\nu(\varphi,z)\),
3. \(\frac{k}{\nu(\varphi,z)}\leq c_{(z,0)}(\varphi_k) \leq \frac{n+k}{\nu(\varphi,z)}\) for any \(z\in\Omega\).
One can take for instance
4. \(\varphi_k(z,w) = \displaystyle\sup_{\xi\in B(z,|w|)}\varphi(\xi)\).
As an application of the above relation between Lelong numbers and complex singularity exponents, the authors present a new proof of Siu’s theorem concerning the analyticity of the set \(\{z:\;\nu(\varphi,z)\geq c\}\), \(c>0\).
1. \(\varphi_k(z,0)=\varphi(z)\),
2. \(\nu(\varphi_k,(z,0))=\nu(\varphi,z)\),
3. \(\frac{k}{\nu(\varphi,z)}\leq c_{(z,0)}(\varphi_k) \leq \frac{n+k}{\nu(\varphi,z)}\) for any \(z\in\Omega\).
One can take for instance
4. \(\varphi_k(z,w) = \displaystyle\sup_{\xi\in B(z,|w|)}\varphi(\xi)\).
As an application of the above relation between Lelong numbers and complex singularity exponents, the authors present a new proof of Siu’s theorem concerning the analyticity of the set \(\{z:\;\nu(\varphi,z)\geq c\}\), \(c>0\).
Reviewer: Fredj Elkhadhra (Kalaa)
MSC:
| 32U25 | Lelong numbers |
| 32U05 | Plurisubharmonic functions and generalizations |
| 32S05 | Local complex singularities |
References:
| [1] | Berndtsson, B., The openness conjecture for plurisubharmonic functions |
| [2] | Bombieri, E., Algebraic values of meromorphic maps, Invent. Math.. Invent. Math., Invent. Math., 11, 163-166 (1970), Addendum: · Zbl 0214.33702 |
| [3] | Demailly, J.-P., Nombres de Lelong généralisés, théorèmes d’intégralité et d’analyticité, Acta Math., 159, 3-4, 153-169 (1987), (in French) · Zbl 0629.32011 |
| [4] | Demailly, J.-P., Multiplier ideal sheaves and analytic methods in algebraic geometry, (School on Vanishing Theorems and Effective Results in Algebraic Geometry. School on Vanishing Theorems and Effective Results in Algebraic Geometry, Trieste, Italy, 2000. School on Vanishing Theorems and Effective Results in Algebraic Geometry. School on Vanishing Theorems and Effective Results in Algebraic Geometry, Trieste, Italy, 2000, ICTP Lect. Notes, vol. 6 (2001), Abdus Salam Int. Cent. Theoret. Phys.: Abdus Salam Int. Cent. Theoret. Phys. Trieste, Italy), 1-148 · Zbl 1102.14300 |
| [5] | Demailly, J.-P., Analytic Methods in Algebraic Geometry (2010), Higher Education Press: Higher Education Press Beijing |
| [6] | Demailly, J.-P., Complex analytic and differential geometry, electronically accessible at: |
| [7] | Demailly, J.-P.; Kollár, J., Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds, Ann. Sci. Éc. Norm. Supér. (4), 34, 4, 525-556 (2001) · Zbl 0994.32021 |
| [8] | Favre, C.; Jonsson, J., The Valuative Tree, Lecture Notes in Mathematics, vol. 1853 (2004), Springer-Verlag: Springer-Verlag Berlin, xiv+234 pp · Zbl 1064.14024 |
| [9] | Favre, C.; Jonsson, M., Valuative analysis of planar plurisubharmonic functions, Invent. Math., 162, 2, 271-311 (2005) · Zbl 1089.32032 |
| [10] | Favre, C.; Jonsson, M., Valuations and multiplier ideals, J. Amer. Math. Soc., 18, 3, 655-684 (2005) · Zbl 1075.14001 |
| [11] | Hörmander, L., An Introduction to Complex Analysis in Several Variables, North-Holland Math. Library, vol. 7 (1990), Elsevier Science Publishers: Elsevier Science Publishers New York: North Holland: Elsevier Science Publishers: Elsevier Science Publishers New York: North Holland Amsterdam · Zbl 0685.32001 |
| [12] | Guan, Q. A.; Zhou, X. Y., Optimal constant in an \(L^2\) extension problem and a proof of a conjecture of Ohsawa, Sci. China Math., 58, 1, 35-59 (2015) · Zbl 1484.32015 |
| [13] | Guan, Q. A.; Zhou, X. Y., A solution of an \(L^2\) extension problem with optimal estimate and applications, Ann. of Math. (2), 181, 3, 1139-1208 (2015) · Zbl 1348.32008 |
| [14] | Guan, Q. A.; Zhou, X. Y., A proof of Demailly’s strong openness conjecture, Ann. of Math. (2), 182, 2, 605-616 (2015) · Zbl 1329.32016 |
| [15] | Guan, Q. A.; Zhou, X. Y., Effectiveness of Demailly’s strong openness conjecture and related problems, Invent. Math., 202, 2, 635-676 (2015) · Zbl 1333.32014 |
| [16] | Guan, Q. A.; Zhou, X. Y., Characterization of multiplier ideal sheaves with weights of Lelong number one, Adv. Math., 285, 1688-1705 (2015) · Zbl 1343.32025 |
| [18] | Kiselman, C. O., Plurisubharmonic functions and potential theory in several complex variables, (Dev. Math. 1950-2000 (2000), Birkhäuser: Birkhäuser Basel, Switzerland), 655-714 · Zbl 0962.31001 |
| [19] | Kollár, J., Flips and abundance for algebraic threefolds, Astérisque, 211 (1992) |
| [20] | Nadel, A., Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Ann. of Math. (2), 132, 3, 549-596 (1990) · Zbl 0731.53063 |
| [21] | Shokurov, V., 3-fold log flips, Izv. Russ. Acad. Nauk Ser. Mat., 56, 105-203 (1992) |
| [22] | Siu, Y. T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., 27, 53-156 (1974) · Zbl 0289.32003 |
| [23] | Siu, Y. T., Multiplier ideal sheaves in complex and algebraic geometry, Sci. China Ser. A, 48, Suppl., 1-31 (2005) · Zbl 1131.32010 |
| [24] | Skoda, H., Sous-ensembles analytiques d’ordre fini ou infini dans \(C^n\), Bull. Soc. Math. France, 100, 353-408 (1972) · Zbl 0246.32009 |
| [25] | Tian, G., On Kähler-Einstein metrics on certain Kähler manifolds with \(C_1(M) > 0\), Invent. Math., 89, 2, 225-246 (1987) · Zbl 0599.53046 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
