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A Survey on the \(L^2\) Extension Theorems

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Results on the extension of \(L^2\) holomorphic functions will be described after a history of the \(L^2\) method. Applications to plurisubharmonic functions and questions in complex geometry will also be reviewed with an emphasis on the Bergman kernel.

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References

  1. Akizuki, Y., Nakano, S.: Note on Kodaira–Spencer’s proof of Lefschetz theorems. Proc. Jpn. Acad. 30, 266–272 (1954)

    Google Scholar 

  2. Andreotti, A., Grauert, H.: Algebraisch Körper von automorphen Funktionen, pp. 39–48. Nachrichten der Akademie der Wiss, Göttingen (1961)

    Google Scholar 

  3. Andreotti, A., Grauert, H.: Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. France 90, 193–259 (1962)

    Google Scholar 

  4. Andreotti, A., Vesentini, E.: Sopra un teorema di Kodaira. Ann. Scuola Norm. Sup. Pisa (3) 15, 283–309 (1961)

    Google Scholar 

  5. Andreotti, A., Vesentini, E.: Carleman estimates for the Laplace–Beltrami equation on complex manifolds. Inst. Hautes Études Sci. Publ. Math. 25, 81–130 (1965)

    Google Scholar 

  6. Angehrn, U., Siu, Y.-T.: Effective freeness and point separation for adjoint bundles. Invent Math 122, 291–308 (1995)

    Google Scholar 

  7. Behnke, H., Thullen, P.: Theorie der Funktionen mehrerer komplexer Veränderlichen, Ergebnisse der Mathematik und ihrer Grenzgebiete. (2. Folge) (by Heinrich Behnke, P. Thullen, R. Remmert, W. Barth, O. Forster, W. Kaup, H. Kerner, H. Holmann, H.J. Reiffen, G. Scheja, K. Spallek) Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge (Book 51) Springer, (1970). (Softcover reprint : October 20, 2011)

  8. Bell, S., Catlin, D.: Boundary regularity of proper holomorphic mappings. Duke Math. J. 49(2), 385–396 (1982)

    Google Scholar 

  9. Bell, S., Ligocka, E.: A simplification and extension of Fefferman’s theorem on biholomorphic mappings. Invent. Math. 57(3), 283–289 (1980)

    Google Scholar 

  10. Bergman, S.: Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande. I. J. Reine Angew. Math. 169, 1–42 (1933)

    Google Scholar 

  11. Berndtsson, B.: The extension theorem of Ohsawa–Takegoshi and the theorem of Donnelly–Fefferman. Ann. Inst. Fourier 46, 1083–1094 (1996)

    Google Scholar 

  12. Berndtsson, B.: Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains. Ann. Inst. Fourier (Grenoble) 56(6), 1633–1662 (2006)

    Google Scholar 

  13. Berndtsson, B.: Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. 169, 531–560 (2009)

    Google Scholar 

  14. Berndtsson, B., Lempert, L.: A proof of Ohsawa–Takegoshi theorem with sharp estimates. J. Math. Soc. Jpn. 68(4), 1461–1472 (2016)

    Google Scholar 

  15. Berndtsson, B., Păun, M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145, 341–378 (2008)

    Google Scholar 

  16. Błocki, Z.: The Bergman kernel and pluripotential theory, Potential theory in Matsue, 1–9. Adv. Stud. Pure Math. 44, 1–9 (2006)

    Google Scholar 

  17. Błocki, Z.: Bergman kernel and metric in terms of logarithmic capacity. Nagoya Math. J. 185, 143–150 (2007)

    Google Scholar 

  18. Błocki, Z.: Suita conjecture and the Ohsawa–Takegoshi extension theorem. Invent. Math. 193, 149–158 (2013)

    Google Scholar 

  19. Błocki, Z., Pflug, P.: Hyperconvexity and Bergman completeness. Nagoya Math. J. 151, 221–225 (1998)

    Google Scholar 

  20. Bombieri, E.: Algebraic values of meromorphic maps. Invent. Math. 10, 267–287 (1970). 9Addendum: Invent. Math. 11, 163–166 (1970)0

    Google Scholar 

  21. Briançon, J., Skoda, H.: Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de \(\mathbb{C}^n,\). C. R. Acad. Sci. Paris Sér. A 278, 949–951 (1974)

    Google Scholar 

  22. Cao, J.-Y.: Ohsawa–Takegoshi Extension Theorem for Compact Kähler Manifolds and Applications, Complex and Symplectic Geometry. Springer INdAM Series, vol. 21, pp. 19–38. Springer, Cham (2017)

    Google Scholar 

  23. Carleson, L.: Selected Problems on Exceptional Sets. Van Nostrand Mathematical Studies, vol. 13, p. v+151. Van Nostrand Co., Inc., Princeton (1967)

    Google Scholar 

  24. Catlin, D.: The Bergman Kernel and a Theorem of Tian, Analysis and Geometry in Several Complex Variables (Katata, 1997), 1–23, Trends Math. Birkhäuser, Boston (1999)

    Google Scholar 

  25. Chen, Boyong: A remark on the Bergman completeness. Complex Var. Theory Appl. 42, 1–15 (2000)

    Google Scholar 

  26. Chen, B.: Boundary behavior of the Bergman metric. Nagoya Math. J. 168, 27–40 (2002)

    Google Scholar 

  27. Chen, B.: A simple proof of the Ohsawa–Takegoshi extension theorem, arXiv:1105.2430

  28. Chow, W.-L.: On compact complex analytic varieties. Am. J. Math. 71, 893–914 (1949)

    Google Scholar 

  29. Claudon, B.: Invariance for multiples of the twisted canonical bundle. Ann. Inst. Fourier (Grenoble) 57, 289–300 (2007)

    Google Scholar 

  30. 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)

    Google Scholar 

  31. Demailly, J.-P.: Cohomology of q-convex spaces in top degrees. Math. Z. 204, 283–295 (1990)

    Google Scholar 

  32. Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361–409 (1992)

    Google Scholar 

  33. Demailly, J.-P.: Analytic Methods in Algebraic Geometry. Higher Education Press, Beijing (2010)

    Google Scholar 

  34. 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)

    Google Scholar 

  35. Diederich, K.: Das Randverhalten der Bergmanschen Kernfunktion und Metrik in streng pseudo-konvexen Gebieten. Math. Ann. 187, 9–36 (1970)

    Google Scholar 

  36. Diederich, K., Herbort, G., Ohsawa, T.: The Bergman kernel on uniformly extendable pseudoconvex domains. Math. Ann. 273(3), 471–478 (1986)

    Google Scholar 

  37. Diederich, K., Pflug, P.: Über Gebiete mit vollständiger Kählermetrik. Math. Ann. 257(2), 191–198 (1981)

    Google Scholar 

  38. Dong, X.: Equality in Suita’s conjecture. arXiv:1807.05537

  39. Donnelly, H., Fefferman, C.: \(L^2\)-cohomology and index theorem for the Bergman metric. Ann. Math. 118, 593–618 (1983)

    Google Scholar 

  40. Donnelly, H., Xavier, F.: On the differential form spectrum of negatively curved Riemannian manifolds. Am. J. Math. 106(1), 169–185 (1984)

    Google Scholar 

  41. Fefferman, C.: On the Bergman kernel and biholomorphic mappings of pseudoconvex domains. Bull. Am. Math. Soc. 80, 667–669 (1974)

    Google Scholar 

  42. Fefferman, C.: Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains. Ann. Math. 103, 395–416 (1976)

    Google Scholar 

  43. Fefferman, C.: Parabolic invariant theory in complex analysis. Adv. Math. 31(2), 131–262 (1979)

    Google Scholar 

  44. Fujita, T.: On Kähler fiber spaces over curves. J. Math. Soc. Japan 30, 779–794 (1978)

    Google Scholar 

  45. Gaffney, M.P.: The harmonic operator for exterior differential forms. Proc. Natl. Acad. Sci. USA 37, 48–50 (1951)

    Google Scholar 

  46. Gaffney, M.P.: A special Stokes’s theorem for complete Riemannian manifolds. Ann. Math. (2) 60, 140–145 (1954)

    Google Scholar 

  47. Garabedian, P.R., Spencer, D.C.: Complex boundary problems. Trans. Am. Math. Soc. 73, 223–242 (1952)

    Google Scholar 

  48. Grauert, H.: Charakterisierung der Holomorphiegebiete durch die vollständige Kählersche Metrik. Math. Ann. 131, 38–75 (1956)

    Google Scholar 

  49. Grauert, H.: On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. 68, 460–472 (1958)

    Google Scholar 

  50. Griffiths, P.A.: Periods of integrals on algebraic manifolds. I and II, Am. J. Math. 90, 568–626 and 805–865 (1968)

  51. 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)

    Google Scholar 

  52. Guan, Q.-A., Zhou, X.-Y.: A proof of Demailly’s strong openness conjecture. Ann. Math. (2) 182(2), 605–616 (2015)

    Google Scholar 

  53. Guan, Q.-A., Zhou, X.-Y.: Strong openness of multiplier ideal sheaves and optimal \(L^ 2\) extension. Sci. China Math. 60(6), 967–976 (2017)

    Google Scholar 

  54. 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)

    Google Scholar 

  55. Guan, Q.-A., Zhou, X.-Y., Zhu, L.-F.: 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 

  56. Herbort, G.: The Bergman metric on hyperconvex domains. Math. Z. 232, 183–196 (1999)

    Google Scholar 

  57. Hirachi, K.: Construction of boundary invariants and the logarithmic singularity of the Bergman kernel. Ann. Math. (2) 151(1), 151–191 (2000)

    Google Scholar 

  58. Hörmander, L.: \(L^2\) estimates and existence theorems for the \(\overline{\partial }\) operator. Acta Math. 113, 89–152 (1965)

    Google Scholar 

  59. Hörmander, L.: An Introduction to Complex Analysis in Several Variables, vol. 7, 3rd edn. North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam (1990)

    Google Scholar 

  60. Hörmander, L.: A history of existence theorems for the Cauchy-Riemann complex in \(L^2\) spaces. J. Geom. Anal. 13, 329–357 (2003)

    Google Scholar 

  61. Hörmander, L., Wermer, J.: Uniform approximation on compact sets in \(\mathbb{C}^n\). Math. Scand. 23, 5–21 (1968)

    Google Scholar 

  62. Hössjer, G.: Über die konforme Abbildung eines Veränderlichen Bereiches. Trans. Chalmers Univ. Technol. Gothenburg Sweden 10, 2–15 (1942)

    Google Scholar 

  63. Hochster, M., Huneke, C.: Tight closure, invariant theory, and the Briançon-Skoda’s theorem. J. Am. Math. Soc. 3(1), 31–116 (1990)

    Google Scholar 

  64. Kai, C., Ohsawa, T.: A note on the Bergman metric of bounded homogeneous domains. Nagoya Math. J. 186, 157–163 (2007)

    Google Scholar 

  65. Kobayashi, S.: Geometry of bounded domains. Trans. Am. Math. Soc. 92, 267–290 (1959)

    Google Scholar 

  66. Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Publications of the Mathematical Society of Japan, Kanô Memorial Lectures, vol. 5, p. xii+305. Princeton University Press, Princeton (1987)

    Google Scholar 

  67. Kodaira, K.: On a differential-geometric method in the theory of analytic stacks. Proc. Natl. Acad. Sci. USA 39, 1268–1273 (1953)

    Google Scholar 

  68. Kodaira, K.: On Kähler varieties of restricted type. Ann. Math. 60, 28–48 (1954)

    Google Scholar 

  69. Kodaira, K., Spencer, D.C.: On deformations of complex analytic structures, I-II. Ann. Math. 67, 328–466 (1958)

    Google Scholar 

  70. Kohn, J.J.: Harmonic integrals on strongly pseudo-convex manifolds. I. Ann. Math. (2) 78, 112–148 (1963)

    Google Scholar 

  71. Kohn, J.J.: Harmonic integrals on strongly pseudo-convex manifolds. II. Ann. Math. (2) 79, 450–472 (1964)

    Google Scholar 

  72. Kohn, J., Nirenberg, L.: Non-coercive boundary value problems. Commun. Pure Appl. Math. 18, 443–492 (1965)

    Google Scholar 

  73. Lipman, J., Teissier, B.: Pseudo-rational local rings and a theorem of Briançon-Skoda. Michgan Math. J. 28, 97–115 (1981)

    Google Scholar 

  74. Maitani, F., Yamaguchi, H.: Variation of Bergman metrics on Riemann surfaces. Math. Ann. 330, 477–489 (2004)

    Google Scholar 

  75. Morrey Jr., C.B.: The analytic embedding of abstract real-analytic manifolds. Ann. Math. 68, 159–201 (1958)

    Google Scholar 

  76. Nakano, S.: On complex analytic vector bundles. J. Math. Soc. Japan 7, 1–12 (1955)

    Google Scholar 

  77. Nakano, S.: Vanishing theorems for weakly 1-complete manifolds, Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya, Tokyo, pp. 169–179 (1973)

  78. Nakano, S., Rhai, T.-S.: Vector bundle version of Ohsawa’s finiteness theorems. Math. Jpn. 24, 657–664 (1979/1980)

  79. Ohsawa, T.: Finiteness theorems on weakly 1-complete manifolds. Publ. Res. Inst. Math. Sci. 15(3), 853–870 (1979)

    Google Scholar 

  80. Ohsawa, T.: On complete Kähler domains with \(C^1\)-boundary. Publ. Res. Inst. Math. Sci. 16(3), 929–940 (1980)

    Google Scholar 

  81. Ohsawa, T.: A reduction theorem for cohomology groups of very strongly q-convex Kähler manifolds. Invent. Math. 63(2), 335–354 (1981)

    Google Scholar 

  82. Ohsawa, T.: On \(H^{ p, q} (X, B)\) of weakly 1-complete manifolds. Publ. Res. Inst. Math. Sci. 17(1), 113–126 (1981)

    Google Scholar 

  83. Ohsawa, T.: Isomorphism theorems for cohomology groups of weakly 1-complete manifolds. Publ. Res. Inst. Math. Sci. 18(1), 191–232 (1982)

    Google Scholar 

  84. Ohsawa, T.: Vanishing theorems on complete Kähler manifolds. Publ. Res. Inst. Math. Sci. 20(1), 21–38 (1984)

    Google Scholar 

  85. Ohsawa, T.: Boundary behavior of the Bergman kernel function on pseudoconvex domains. Publ. Res. Inst. Math. Sci. 20(5), 897–902 (1984)

    Google Scholar 

  86. Ohsawa, T.: On the Bergman kernel of hyperconvex domains. Nagoya Math. J. 129, 43–52 (1993)

    Google Scholar 

  87. Ohsawa, T.: On the Extension of \(L^2\) Holomorphic Functions. IV. A New Density Concept., Geometry and Analysis on Complex Manifolds, pp. 157–170. World Scientific Publishing, River Edge (1994)

    Google Scholar 

  88. Ohsawa, T.: Addendum to: “On the Bergman kernel of hyperconvex domains” [Nagoya Math. J. 129, 43–52 (1993); MR1210002]. Nagoya Math. J. 137, 145–148 (1995)

    Google Scholar 

  89. Ohsawa, T.: On the extension of \(L^2\) holomorphic functions. III. Negligible weights. Math. Z. 219(2), 215–225 (1995)

    Google Scholar 

  90. Ohsawa, T.: Bergman kernel and the Suita conjecture on Riemann surfaces, (Japanese) The theory of reproducing kernels and their applications (Japanese) (Kyoto, 1998). \({\rm S\overline{u}rikaisekikenky\overline{u}sho\; K\overline{o}ky\overline{u}roku}\) No. 1067, 89–95 (1998)

  91. Ohsawa, T.: On the extension of \(L^2\) holomorphic functions VIII—a remark on a theorem of Guan and Zhou. Int. J. Math. 28(9), 1740005 (2017). 12 pp

    Google Scholar 

  92. Ohsawa, T.: \(L^2\) Approaches in Several Complex Variables—Towards the Oka-Cartan Theory with Precise Bounds. Springer Monographs in Mathematics, 2nd edn, p. 258. Springer, Tokyo (2018)

    Google Scholar 

  93. Ohsawa, T., Takegoshi, K.: On the extension of \(L^ 2\) holomorphic functions. Math. Z. 195(2), 197–204 (1987)

    Google Scholar 

  94. Ohsawa, T.: Hodge spectral sequence on pseudoconvex domains. Math. Z. 197(1), 1–12 (1988)

    Google Scholar 

  95. Oka, K.: Sur les fonctions analytiques de plusieurs variables. IV. Domaines pseudoconvexes. Tôhoku Math. J. 49, 15–52 (1942)

    Google Scholar 

  96. Oka, K.: Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques. Bull. Soc. Math. France 78, 1–27 (1950)

    Google Scholar 

  97. Oka, K.: Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur. Jap. J. Math. 23, 97–155 (1954)

    Google Scholar 

  98. Păun, M.: Siu’s invariance of plurigenera: a one-tower proof. J. Differ. Geom. 76, 485–493 (2007)

    Google Scholar 

  99. Păun, M., Takayama, S.: Positivity of twisted relative pluricanonical bundles and their direct images. J. Algebraic Geom. 27, 211–272 (2018)

    Google Scholar 

  100. Pflug, P.: Quadratintegrable holomorphe Funktionen und die Serre-Vermutung. Math. Ann. 216, 285–288 (1975)

    Google Scholar 

  101. Pflug, P.: Holomorphiegebiete, pseudokonvexe Gebiete und das Levi-Problem. Lecture Notes in Mathematics, vol. 432. Springer-Verlag, Berlin (1975)

    Google Scholar 

  102. Pflug, P.: Various applications of the existence of well growing holomorphic functions, Functional analysis, holomorphy and approximation theory (Rio de Janeiro, 1980), pp. 391–412, North-Holland Math. Stud., 71, Notas Mat., 88, North-Holland, Amsterdam, (1982)

  103. Sario, L., Oikawa, K.: Capacity functions, Die Grundlehren der mathematischen Wissenschaften, Band 149, p. xvii+361. Springer-Verlag, New York (1969)

    Google Scholar 

  104. Sario, L., Oikawa, K.: Beurling type density theorems in the unit disk. Invent. Math. 113, 21–39 (1993)

    Google Scholar 

  105. Schiffer, M.M.: Variational methods in the theory of conformal mapping. Proceedings of the International Congress of Mathematicians, Cambridge, Masschussetts, vol. 2, pp. 233–240. American Mathematical Society, Providence, R.I. (1952)

  106. Siciak, J.: On removable singularities of \(L^2\) holomorphic functions of several variables, Prace matematyczno-fizyezne Wyzsza Szkota Inzynierska w Radomiu, pp. 73–81 (1982)

  107. Siegel, C.: Analytic functions of several complex variables, Lectures delivered at the Institute for Advanced Study, 1948–1949. With notes by P. T. Bateman. Reprint of the 1950 edition. Kendrick Press, Heber City, UT, (2008). viii+174 pp

  108. Siegel, C.: Topics in complex function theory. Vol. III. Abelian functions and modular functions of several variables, Translated from the German by E. Gottschling and M. Tretkoff. With a preface by Wilhelm Magnus. Reprint of the 1973 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, x+244 pp (1989)

  109. Siu, Y.-T.: Invariance of plurigenera. Invent. Math. 134, 661–673 (1998)

    Google Scholar 

  110. Siu, Y.-T.: Extension of Twisted Pluricanonical Sections with Plurisubharmonic Weight and Invariance of Semipositively Twisted Plurigenera for Manifolds not Necessarily of General type, Complex Geometry (Göttingen, 2000), 223–277. Springer, Berlin (2002)

    Google Scholar 

  111. Siu, Y.-T., Yau, S.-T.: Compactification of Negatively Curved Complete Kähler Manifolds of Finite Volume. Seminar on Differential Geometry, vol. 10, pp. 363–380. Princeton University Press, Princeton (1982)

    Google Scholar 

  112. Skoda, H.: Application des techniques \(L^2\) à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids. Ann. Sci. École Norm. Sup. 5, 545–579 (1972)

    Google Scholar 

  113. Suita, N.: Capacities and kernels on Riemann surfaces. Arch. Ration. Mech. Anal. 46, 212–217 (1972)

    Google Scholar 

  114. Suita, N.: On conformal invariants (Japanese), https://doi.org/10.11429/emath1996.1998.Autumn-Meeting1_48

  115. Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Differ. Geom. 32, 99–130 (1990)

    Google Scholar 

  116. Zelditch, S.: Szegö kernels and a theorem of Tian. Int. Math. Res. Notices 6, 317–331 (1998)

    Google Scholar 

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  1. Graduate School of Mathematics, Nagoya University, Chikusaku Furocho, Nagoya, 464-8602, Japan

    Takeo Ohsawa

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The author expresses his deep gratitude to M. Jarnicki and P. Pflug for giving him an opportunity of writing this survey. He also thanks to S. Fu for recommending to read [60] in July 2003 in Nagoya. Last but not least, he thanks to the referee for useful suggestions.

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Ohsawa, T. A Survey on the \(L^2\) Extension Theorems. J Geom Anal 30, 1366–1395 (2020). https://doi.org/10.1007/s12220-019-00349-2

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