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A Survey on L2 Extension Problem

  • Xiangyu ZhouEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 10)

Abstract

In the present paper, we’ll give a survey of our recent results on the L2 extension problem with optimal estimate. We’ll consider the problem in various settings according to Ohsawa’s series papers, and present our optimal versions of Ohsawa’s L2 extension theorems. We’ll discuss the problem in a general setting and present a solution of the problem in the general setting. We’ll give some applications of our results including a solution of the equality part of Suita’s conjecture. Finally, we present our recent solutions of Demailly’s strong openness conjecture on multiplier ideal sheaf and related problems.

Keywords

Optimal Estimate Holomorphic Section Bergman Kernel Holomorphic Vector Bundle Stein Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Berndtsson, B.: The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman. Ann. L’Inst. Fourier (Grenoble) 46(4), 1083–1094 (1996)Google Scholar
  2. 2.
    Berndtsson, B.: Prekopa’s theorem and Kiselman’s minimal principle for plurisubharmonic functions. Math. Ann. 312, 785–92 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Berndtsson, B.: Integral formulas and the Ohsawa-Takegoshi extension theorem. Sci. China Ser. A 48(suppl), 61–73 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Berndtsson, B.: Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann. Inst. Fourier (Grenoble) 56, 1633–1662 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Berndtsson, B.: Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. 169, 531–560 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Berndtsson, B., Păun, M.: A Bergman kernel proof of the Kawamata subadjunction theorem (2008). arXiv:0804.3884v2Google Scholar
  7. 7.
    Berndtsson, B., Păun, M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke. Math. J. 145, 341–378 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Berndtsson, B., Păun, M.: Bergman kernels and subadjunction (2010). arXiv:1002.4145v1Google Scholar
  9. 9.
    Berndtsson, B., Păun, M.: Quantitative extensions of pluricanonical forms and closed positive currents. Nagoya Math. J. 205, 25–65 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Berndtsson, B.: The openness conjecture for plurisubharmonic functions (2013). arXiv:1305.5781Google Scholar
  11. 11.
    Blocki, Z.: On the Ohsawa-Takegoshi extension theorem. Univ. Iag. Acta Math. 50, 53–61 (2012)MathSciNetGoogle Scholar
  12. 12.
    Blocki, Z.: Suita conjecture and the Ohsawa-Takegoshi extension theorem. Invent Math. 193, 149–158 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Boucksom, S., Favre, C., Jonsson, M.: Valuations and plurisubharmonic singularities. Publ. Res. Inst. Math. Sci. 44(2), 449–494 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Cao, J.Y.: Numerical dimension and a Kawamata-Viehweg-Nadel type vanishing theorem on compact Kähler manifolds (2012). arXiv:1210.5692.Google Scholar
  15. 15.
    Cao, J., Shaw, M.-C., Wang, L.: Estimates for the \(\bar{\partial }\)-Neumann problem and nonexistence of C 2 Levi-flat hypersurfaces in CP n. Math. Z. 248, 183–221 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Chen, S.-C., Shaw, M.-C.: Partial Differential Equations in Several Complex Variables. AMS/IP, Providence (2001)zbMATHGoogle Scholar
  17. 17.
    Demailly, J-P.: Multiplier ideal sheaves and analytic methods in algebraic geometry. In: School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000). ICTP Lecture Notes, vol. 6, pp. 1–148. Abdus Salam International Centre for Theoretical Physics, Trieste (2001)Google Scholar
  18. 18.
    Demailly, J.-P.: Estimations L 2 pour l’opérateur d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète. (French) [L 2-estimates for the \(\bar{\partial }\)-operator of a semipositiv holomorphic vector bundle over a complete Kähler manifold] Ann. Sci. école Norm. Sup. (4) 15(3), 457–511 (1982)Google Scholar
  19. 19.
    Demailly, J.-P.: On the Ohsawa-Takegoshi-Manivel L 2 extension theorem. In: Proceedings of the Conference in Honour of the 85th Birthday of Pierre Lelong, Paris, Sept 1997. Progress in Mathematics. Birkhäuser, Basel (2000)Google Scholar
  20. 20.
    Demailly, J.-P.: Complex analytic and differential geometry (2012), electronically accessible at http://www-fourier.ujf-grenoble.fr/~demailly/books.html
  21. 21.
    Demailly, J.-P.: Analytic Methods in Algebraic Geometry. Higher Education Press, Beijing (2010)Google Scholar
  22. 22.
    Demailly, J.-P., Hacon, C.D., Păun, M.: Extension theorems, non-vanishing and the existence of good minimal models. Acta Math. 210(2), 203–259 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Demailly, J-P., Ein, L., Lazarsfeld, R.: A subadditivity property of multiplier ideals. Mich. Math. J. 48, 137–156 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Demailly, J-P., Kollár, J.: Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Sci. École Norm. Sup. (4) 34(4), 525–556 (2001)Google Scholar
  25. 25.
    Demailly, J-P., Peternell, T.: A Kawamata-Viehweg vanishing theorem on compact Kähler manifolds. J. Differ. Geom. 63(2), 231–277 (2003)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Demailly, J.P.: Kähler manifolds and transcendental techniques in algebraic geometry. In: Proceedings of the ICM, Zurich. European Mathematical Society, vol. 1, pp. 153–186 (2007)MathSciNetGoogle Scholar
  27. 27.
    Diederich, K., Herbort, G.: Extension of holomorphic L 2 functions with weighted growth conditions. Nagaya Math. J. 126, 144–157 (1992)MathSciNetGoogle Scholar
  28. 28.
    Dinh, T.-C., Sibony, N.: Super-potentials of postive closed currents, intersection theory and dynamics. Acta Math. 203, 1–82 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Duval, J., Sibony, N.: Polynomial convexity, rational convexity, and currents. Duke Math. J. 79(2), 487–513 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Favre, C., Jonsson, M.: Valuative analysis of planar plurisubharmonic functions. Invent. Math. 162(2), 271–311 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Favre, C., Jonsson, M.: Valuations and multiplier ideals. J. Am. Math. Soc. 18(3), 655–684 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Fornæss J.E., Narasimhan, R.: The Levi problem on complex spaces with singularities. Math. Ann. 248(1), 47–72 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Fornaess, J.E., Sibony, N.: Some open problems in higher dimensional complex analysis and complex dynamics. Publ. Mat. 45(2), 529–547 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Fornaess, J.E., Stensønes, B.: Lectures on Counterexamples in Several Complex Variables. AMS, Providence (2007)zbMATHGoogle Scholar
  35. 35.
    Farkas, H.M., Kra, I.: Riemann surfaces. In: Graduate Texts in Mathematics, vol. 71. Springer, New York/Berlin (1980)Google Scholar
  36. 36.
    Grauert, H.: Selected Papers. Springer, Berlin/New York (1994)zbMATHGoogle Scholar
  37. 37.
    Grauert, H., Remmert, R.: Coherent analytic sheaves. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265. Springer, Berlin (1984)Google Scholar
  38. 38.
    Grauert, H., Remmert, R.: Theory of Stein Spaces. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 236. Springer, Berlin (1979)Google Scholar
  39. 39.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Pure and Applied Mathematics, pp. xii+813. Wiley-Interscience, New York (1978). ISBN: 0-471-32792-1Google Scholar
  40. 40.
    Guan, Q.A., Zhou, X.Y., Zhu, L.F.: On the Ohsawa-Takegoshi L 2 extension theorem and the twisted Bochner-Kodaira identity. C. R. Math. Acad. Sci. Paris 349(13–14), 797–800 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Guan, Q.A., Zhou, X.Y.: Optimal constant problem in the L 2 extension theorem. C. R. Math. Acad. Sci. Paris 350(15–16), 753–756 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Guan, Q.A., Zhou, X.Y.: Generalized L 2 extension theorem and a conjecture of Ohsawa. C. R. Math. Acad. Sci. Paris 351(3–4), 111–114 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Guan, Q.A., Zhou, X.Y.: Optimal constant in L 2 extension and a proof of a conjecture of Ohsawa. Sci. China Math. 58(1), 35–59 (2015)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Guan, Q.A., Zhou, X.Y.: An L 2 extension theorem with optimal estimate. C. R. Acad. Sci. Paris, Ser. I 352(2), 137–141 (2014)Google Scholar
  45. 45.
    Guan, Q.A., Zhou, X.Y.: A solution of an L 2 extension problem with optimal estimate and applications. Ann. Math. (2014). arXiv:1310.7169; published onlineGoogle Scholar
  46. 46.
    Guan, Q.A., Zhou, X.Y.: Strong openness conjecture for plurisubharmonic functions (2013). arXiv:1311.3781Google Scholar
  47. 47.
    Guan, Q.A., Zhou, X.Y.: Strong openness conjecture and related problems for plurisubharmonic functions (2014). arXiv:1401.7158Google Scholar
  48. 48.
    Guan, Q.A., Zhou, X.Y.: Effectiveness of Demailly’s strong openness conjecture and related problems (2014). arXiv:1403.7247; accepted by Invent. Math.Google Scholar
  49. 49.
    Gunning, R.C., Rossi, H.: Analytic Functions of Several Complex Variables. Prentice-Hall, Englewood Cliffs (1965)zbMATHGoogle Scholar
  50. 50.
    Hörmander, L.: L 2 estimates and existence theorems for the \(\overline{\partial }\) operater. Acta Math. 113, 89–152 (1965)zbMATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators II. Grundlehren der Mathematischen Wissenschaften, vol. 257. Springer, Berlin (1983)Google Scholar
  52. 52.
    Hömander, L.: An Introduction to Complex Analysis in Several Variables, 3rd edn. Elsevier Science, Amsterdam/New York (1990)Google Scholar
  53. 53.
    Jonsson, M., Mustată, M.: Valuations and asymptotic invariants for sequences of ideals. Annales de l’Institut Fourier A. 62(6), 2145–2209 (2012)zbMATHCrossRefGoogle Scholar
  54. 54.
    Jonsson, M., Mustată, M.: An algebraic approach to the openness conjecture of Demailly and Kollár. J. Inst. Math. Jussieu 13(1), 119–144 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Kiselman, C.O.: Plurisubharmonic functions and potential theory in several complex variables. In: Development of Mathematics 1950–2000, pp 654–714. Birkhäuser, Basel (2000)Google Scholar
  56. 56.
    Manivel, L.: Un théorème de prolongement L 2 de sections holomorphes d’un fibré vectoriel. Math. Z. 212, 107–122 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    McNeal, J., Varolin, D.: Analytic inversion of adjunction: L 2 extension theorems with gain. Ann. L’Inst. Fourier (Grenoble) 57(3), 703–718 (2007)Google Scholar
  58. 58.
    Nadel, A.: Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. Math. (2) 132(3), 549–596 (1990)Google Scholar
  59. 59.
    Ohsawa, T.: On the extension of L 2 holomorphic functions. II. Publ. Res. Inst. Math. Sci. 24(2), 265–275 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    Ohsawa, T.: On the extension of L 2 holomorphic functions. III. Negligible weights. Math. Z. 219(2), 215–225 (1995)zbMATHMathSciNetGoogle Scholar
  61. 61.
    Ohsawa, T.: On the extension of L 2 holomorphic functions. IV. A new density concept. In: Geometry and Analysis on Complex Manifolds, pp. 157–170. World Scientific, River Edge (1994)Google Scholar
  62. 62.
    Ohsawa, T.: Addendum to “On the Bergman kernel of hyperconvex domains”. Nagoya Math. J. 137, 145–148 (1995)zbMATHMathSciNetGoogle Scholar
  63. 63.
    Ohsawa, T.: On the extension of L 2 holomorphic functions. V. Effects of generalization. Nagoya Math. J. 161, 1–21 (2001). Erratum to: “On the extension of L 2 holomorphic functions. V. Effects of generalization” [Nagoya Math. J. 161, 1–21 (2001)]. Nagoya Math. J. 163, 229 (2001)Google Scholar
  64. 64.
    Ohsawa, T.: On the extension of L 2 holomorphic functions. VI. a limiting case. Contemp. Math. 332, 235–239 (2003)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Ohsawa, T.: Application of a sharp L 2 extension theorem (2013, preprint)Google Scholar
  66. 66.
    Ohsawa, T., Takegoshi, K.: On the extension of L 2 holomorphic functions. Math. Z. 195, 197–204 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Păun, M.: Siu’s invariance of plurigenera: a one-tower proof. J. Differ. Geom. 76(3), 485–493 (2007)zbMATHGoogle Scholar
  68. 68.
    Pommerenke, C., Suita, N.: Capacities and Bergman kernels for Riemann surfaces and Fuchsian groups. J. Math. Soc. Jpn. 36(4), 637–642 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    Sario, L., Oikawa, K.: Capacity Functions. Die Grundlehren der Mathematischen Wissenschaften, Band 149. Springer, New York (1969)Google Scholar
  70. 70.
    Sergeev, A., Zhou, X.Y.: Invariant domains of holomorphy: twenty years later. Proc. of the Steklov Math. Inst. 285, 241–250 (2014)zbMATHCrossRefGoogle Scholar
  71. 71.
    Sibony, N.: Quelques problèmes de prolongement de courants en analyse complexe. (French) [Some extension problems for currents in complex analysis] Duke Math. J. 52(1), 157–197 (1985)Google Scholar
  72. 72.
    Sibony, N.: Some recent results on weakly pseudoconvex domains. In: Proceedings of the ICM, Tokyo. Mathematical Society, pp. 943–950 (1991)Google Scholar
  73. 73.
    Siu, Y.T.: Pseudoconvexity and the problem of Levi. Bull. AMS. 84(4), 481–512 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  75. 75.
    Siu, Y.-T.: The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi. In: Geometric Complex Analysis, Hayama, pp. 577–592. World Scientific, Singapore/New Jersey (1996)Google Scholar
  76. 76.
    Siu, Y.-T.: Invariance of plurigenera. Invent. Math. 134(3), 661–673 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  77. 77.
    Siu, Y.-T.: Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. In: Complex Geometry (Göttingen, 2000), pp. 223–277. Springer, Berlin (2002)Google Scholar
  78. 78.
    Siu, Y.T.: Invariance of plurigenera and torsion-freeness of direct image sheaves of pluricanonical bundles. In: Finite or Infinite Dimensional Complex Analysis and Applications, pp. 45–83. Kluwer Academic, Dordrecht (2004)Google Scholar
  79. 79.
    Siu, Y.-T.: Some recent transcendental techniques in algebraic and complex geometry. In: Proceedings of the ICM, vol. 1, pp. 439–448. Higher Educational Press, Beijing (2002)Google Scholar
  80. 80.
    Siu, Y.T.: Multiplier ideal sheaves in complex and algebraic geometry. Sci. China Ser. A 48(suppl), 1–31 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  81. 81.
    Siu, Y.T.: Dynamic multiplier ideal sheaves and the construction of rational curves in Fano manifolds. In: Complex Analysis and Digital Geometry. Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., vol. 86, pp. 323–360. Uppsala Universitet, Uppsala (2009)Google Scholar
  82. 82.
    Siu, Y.T.: Function theory of several complex variables. In: Chineses, Notes by Chen Zhihua and Zhong Jiaqing. Higher Educational Press, Beijing (2013)Google Scholar
  83. 83.
    Straube, E.: Lectures on the L 2-Sobolev Theory of the \(\bar{\partial }\)-Neumann Problem. ESI Lectures in Mathematics and Physics. European Mathematical Society, Zürich (2010)zbMATHGoogle Scholar
  84. 84.
    Suita, N.: Capacities and kernels on Riemann surfaces. Arch. Ration. Mech. Anal. 46, 212–217 (1972)zbMATHMathSciNetCrossRefGoogle Scholar
  85. 85.
    Suita, N., Yamada, A.: On the Lu Qi-keng conjecture. Proc. Am. Math. Soc. 59(2), 222–224 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    Yamada, A.: Topics related to reproducing kemels, theta functions and the Suita conjecture (Japanese). The theory of reproducing kemels and their applications (Japanese) (Kyoto, 1998). Su-rikaisekikenkyu-sho Ko-kyu-roku No. 1067, 39–47 (1998)Google Scholar
  87. 87.
    Zhou, X.: Some results related to group actions in several complex variables. In: Proceedings of the International Congress of Mathematicians, Beijing, vol. II, pp. 743–753. Higher Education Press, Beijing (2002)Google Scholar
  88. 88.
    Zhou, X.: Invariant holomorphic extensionin in several complex variables. Sci. China Ser. A, 49(11), 1593–1598 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  89. 89.
    Zhou, X.Y., Zhu, L.F.: L 2-extension theorem: revisited. In: Fifth International Congress of Chinese Mathematicians. Part 1, 2. AMS/IP Studies in Advanced Mathematics, vol. 51, pp. 475–490, pt. 1, 2. American Mathematical Society, Providence (2012)Google Scholar
  90. 90.
    Zhou, X.Y., Zhu, L.F.: L 2 extension with optimal estimate on weakly pseudoconvex manifolds (Preprint, 2013)Google Scholar
  91. 91.
    Zhu, L.F., Guan, Q.A., Zhou, X.Y.: On the Ohsawa-Takegoshi L 2 extension theorem and the Bochner-Kodaira identity with non-smooth twist factor. J. Math. Pures Appl. (9) 97(6), 579–601 (2012)Google Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Yau Mathematical Center, Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingChina

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