A Role of the L2 Method in the Study of Analytic Families

  • Takeo OhsawaEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 309)


An expository account is given on the \(L^2\) method for the \(\bar{\partial }\) equation, \(L^2\) extension theorems and the Bergman kernel focusing on the recent applications to analytic families.


Analytic family 

2010 Mathematics Subject Classification

Primary 32E40 Secondary 32T05 


  1. 1.
    Andreotti, A., Vesentini, E.: Sopra un teorema di Kodaira. Ann. Scuola Norm. Sup. Pisa (3) 15, 283–309 (1961)Google Scholar
  2. 2.
    Andreotti, A., Vesentini, E.: Carleman estimates for the Laplace–Beltrami equation on complex manifolds. Inst. Hautes Études Sci. Publ. Math. 25, 81–130 (1965)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berndtsson, B.: Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann. Inst. Fourier (Grenoble) 56, 1633–1662 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berndtsson, B.: Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. 169, 531–560 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Berndtsson, B., Lempert, L.: A proof of the Ohsawa–kegoshi theorem with sharp estimates. J. Math. Soc. Japan 68(4), 1461–1472 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Błocki, Z.: Suita conjecture and the Ohsawa–Takegoshi extension theorem. Invent. Math. 193, 149–158 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Błocki, Z.: Bergman kernel and pluripotential theory, In: Analysis, Complex Geometry, and Mathematical Physics: In Honor of Duong H. Phong, 1–10, Contemporary Mathematics, vol. 644. American Mathematical Society, Providence, RI (2015)Google Scholar
  8. 8.
    Bremermann, H.J.: Die Charakterisierung von Regülaritätsgebieten durch pseudokonvexe Funktionen. Schr. Math. Inst. Univ. Münster, 5, i+92 (1951)Google Scholar
  9. 9.
    Bremermann, H.J.: On the conjecture of the equivalence of the plurisubharmonic functions and the Hartogs functions. Math. Ann. 131, 76–86 (1956)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cao, J.-Y.: Numerical dimension and a Kawamata-Viehweg-Nadel-type vanishing theorems on compact Kähler manifolds. Compos. Math. 150, 1869–1902 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chirka, E.M.: Holomorphic motions and the uniformization of holomorphic families of Riemann surfaces, (Russian) Uspekhi Mat. Nauk 67 (2012), 6(408), 125–202 (Translation in Russian Math. Surveys 67(6), 1091–1165 (2012))Google Scholar
  12. 12.
    Demailly, J.-P.: Estimations \(L^2\) pour l’opérateur \({\bar{\partial }}\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète. Ann. Sci. École Norm. Sup. 15, 457–511 (1982)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chirka, E.M.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361–409 (1992)MathSciNetGoogle Scholar
  14. 14.
    Chirka, E.M.: Analytic Methods in Algebraic Geometry. Higher Education Press, Beijing (2010)Google Scholar
  15. 15.
    Deng, F.-S., Wang, Z.-W., Zhang, L.-Y., Zhou, X.-Y.: New characterizations of plurisubharmonic functions and positivity of direct image sheaves. arXiv:1809.10371
  16. 16.
    Dong, R.X.: Equality in Suita’s conjecture. arXiv:1807.05537v1
  17. 17.
    Fefferman, C.: Monge–Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math. 103, 395–416 (1976)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fischer, W., Grauert, H.: Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten, Nachr. Akad. Wiss. Gttingen Math.-Phys. Kl. II, 89-94 (1965)Google Scholar
  19. 19.
    Furushima, M., Nobe, M., Ohshima, Y.: A note on minimal normal compactifications of \(\mathbb{C}^2\). Kumamoto J. Math. 27, 5–21 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Guan, Q.-A., Zhou, X.-Y.: A solution of an \(L^2\) extension problem with optimal estimate and applications. Ann. Math. 181, 1139–1208 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Guan, Q.-A., Zhou, X.-Y.: Charakterisierung der Holomorphiegebiete durch die vollständige Kählersche Metrik. Math. Ann. 131, 38–75 (1956)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Guan, Q.-A., Zhou, X.-Y.: On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. 68, 460–472 (1958)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hartogs, F.: Über die aus den singulären Stellen einer analytishcen Funktion mehrerer Veränderlichen bestehenden Gebilde. Acta Math. 32, 57–79 (1909)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hörmander, L.: \(L^2\) estimates and existence theorems for the \(\overline{\partial }\) operator. Acta Math. 113, 89–152 (1965)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hössjer, G.: Über die konforme Abbildungeines Veränderlichen Bereiches (Transactions of Chalmers University of Technology Gothenburg, Sweden) vol. 10, pp. 2–15 (1942)Google Scholar
  26. 26.
    Hosono, G.: The optimal jet \(L^2\) extension of Ohsawa-Takegoshi type, arXiv:1706.08725 [math.CV]
  27. 27.
    Kodaira, K.: On a differential-geometric method in the theory of analytic stacks. Proc. Nat. Acad. Sci. U.S.A. 39, 1268–1273 (1953)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kodaira, K.: On Kähler varieties of restricted type. Ann. Math. 60, 28–48 (1954)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kodaira, K.: Holomorphic mappings of polydiscs into compact complex manifolds. J. Differ. Geom. 6, 33–46 (1971/1972)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kodaira, K., Spencer, D.C.: On deformations of complex analytic structures, I-II. Ann. Math. 67, 328–466 (1958)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Maitani, F.: Variations of meromorphic differentials under quasiconformal deformations. J. Math. Kyoto Univ. 24, 49–66 (1984)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Maitani, F., Yamaguchi, H.: Variation of Bergman metrics on Riemann surfaces. Math. Ann. 330, 477–489 (2004)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Matsumura, S.: A vanishing theorem of Kollár–Ohsawa type. Math. Ann. 366, 1451–1465 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Morrow, J.: Minimal normal compactifications of \({\mathbb{C}}^{*})^2\), Complex analysis, 1972. In: Proceedings of the Conference Rice University, Houston, Texas, 1972, Vol. I. Geometry of singularities. Rice University Studies, vol. 59, Issue 1, pp. 97–112 (1973)Google Scholar
  35. 35.
    Nadel, A.: Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. Math. 132, 549–596 (1990)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Nishino, T.: Nouvelles recherches sur les fonctions entières de plusieurs variables complexes. I. J. Math. Kyoto Univ. 8, 49–100 (1968)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Nishino, T.: Nouvelles recherches sur les fonctions entières de plusieurs variables complexes. II. Fonctions entières qui se réduisent à celles d’une variable. J. Math. Kyoto Univ. 9, 221–274 (1969)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Nishino, T.: Nouvelles recherches sur les fonctions entières de plusiers variables complexes. III. Sur quelques propriétés topologiques des surfaces premières. J. Math. Kyoto Univ. 10, 245–271 (1970)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Nishino, T.: Nouvelles recherches sur les fonctions entières de plusieurs variables complexes. IV. Types de surfaces premières. J. Math. Kyoto Univ. 13, 217–272 (1973)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Nishino, T.: Nouvelles recherches sur les fonctions entières de plusieurs variables complexes. V. Fonctions qui se réduisent aux polynômes. J. Math. Kyoto Univ. 15(3), 527–553 (1975)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Nishino, T.: Value distribution of analytic functions of two variables (Japanese). Sūgaku 32(3), 230–246 (1980)Google Scholar
  42. 42.
    Norguet, F.: Sur les domaines d’holomorphie des fonctions uniformes de plusieurs variables complexes (Passage du local au global). Bull. Soc. Math. France 82, 137–159 (1954)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Ohsawa, T.: Analyticity of complements of complete Kähler domains. Proc. Japan Acad. Ser. A Math. Sci. 56, 484–487 (1980)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Ohsawa, T.: On complete Kähler domains with \(C^1\)-boundary. Publ. Res. Inst. Math. Sci. 16(3), 929–940 (1980)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Ohsawa, T.: Vanishing theorems on complete Kähler manifolds. Publ. Res. Inst. Math. Sci. 20(1), 21–38 (1984)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Ohsawa, T.: On the Bergman kernel of hyperconvex domains. Nagoya Math. J. 129, 43–52 (1993). (Addendum, Nagoya Math. J. 137, 145–148 (1995))Google Scholar
  47. 47.
    Ohsawa, T.: On the extension of \(L^2\) holomorphic functions V. Effect of generalization. Nagoya Math. J. 161, 1–21 (2001)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Ohsawa, T.: \(L^2\) proof of Nishino’s rigidity theorem, to appear in Kyoto J. MathGoogle Scholar
  49. 49.
    Ohsawa, T.: Generalizations of theorems of Nishino and Hartogs by the \(L^2\) method, in preparationGoogle Scholar
  50. 50.
    Ohsawa, T., Takegoshi, K.: On the extension of \( L^2\) holomorphic functions. Math. Z. 195, 197–204 (1987)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Oka, K.: Sur les fonctions de plusieurs variables. IX. Domaines finis sans point critique intérieur, Jap. J. Math. 23, 97–155 (1953)Google Scholar
  52. 52.
    Shcherbina, N.: Pluripolar graphs are holomorphic. Acta Math. 194, 203–216 (2005)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Suita, N.: Capacities and kernels on Riemann surfaces. Arch. Ration. Mech. Anal. 46, 212–217 (1972)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Suzuki, M.: Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace \(\mathbb{C}^2\). J. Math. Soc. Jpn. 26, 241–257 (1974)CrossRefGoogle Scholar
  55. 55.
    Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Diff. Geom. 32, 99–130 (1990)CrossRefGoogle Scholar
  56. 56.
    Ueda, T.: Compactifications of \({\mathbb{C}}\times {\mathbb{C}}^{*}\) and \(({\mathbb{C}}^{*})^2\). Tôhoku Math. J. 2 31(1), 81–90 (1979)Google Scholar
  57. 57.
    Yamaguchi, H.: Parabolicité d’une fonction entière. J. Math. Kyoto Univ. 16, 71–92 (1976)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Yamaguchi, H.: Complex and vector potential theory, (Japanese). Sūgaku 50(3), 225–247Google Scholar

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Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan

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