Advertisement

Science China Mathematics

, Volume 58, Issue 1, pp 35–59 | Cite as

Optimal constant in an L 2 extension problem and a proof of a conjecture of Ohsawa

  • Qi’An Guan
  • XiangYu ZhouEmail author
Articles Progress of Projects Supported by NSFC

Abstract

In this paper, we solve the optimal constant problem in the setting of Ohsawa’s generalized L 2 extension theorem. As applications, we prove a conjecture of Ohsawa and the extended Suita conjecture, we also establish some relations between Bergman kernel and logarithmic capacity on compact and open Riemann surfaces.

Keywords

L2 extension theorem optimal L2 estimate Bergman kernel a conjecture of Ohsawa extended Suita conjecture 

MSC(2010)

Primary 32D15, 32E10, 32A25, 32L10, 32U05 Secondary 32W05, 14H55, 32Q28 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arakelov S Yu. Intersection theory of divisors on an arithmetic surface. Izv Akad Nauk SSSR Ser Mat, 1974, 38: 1179–1192MathSciNetGoogle Scholar
  2. 2.
    Berndtsson B. The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman. Ann L’Inst Fourier (Grenoble), 1996, 46: 1083–1094CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Berndtsson B. Integral formulas and the Ohsawa-Takegoshi extension theorem. Sci China Ser A, 2005, 48(supp): 61–73CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Berndtsson B. An introduction to things ∂̄. In: Analytic and Algebraic Geometry. IAS/Park City Math Ser, 17. Providence, RI: Amer Math Soc, 2010, 7–76Google Scholar
  5. 5.
    Blocki Z. On the Ohsawa-Takegoshi extension theorem. Univ Jag Acta Math, 2012, 50: 53–61Google Scholar
  6. 6.
    Blocki Z. Suita conjecture and the Ohsawa-Takegoshi extension theorem. Invent Math, 2013, 193: 149–158CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Cao J, Shaw M-C, Wang L. Estimates for the ∂̄-Neumann problem and nonexistence of C 2 Levi-flat hypersurfaces in CP n. Math Z, 2004, 248: 183–221CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Chen S-C, Shaw M-C. Partial Differential Equations in Several Complex Variables. AMS/IP Studies in Advanced Mathematics, 19. Providence, RI/Boston, MA: Amer Math Soc/International Press, 2001zbMATHGoogle Scholar
  9. 9.
    Demailly J-P. Estimations L 2 pour l’opérateur d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète (in French) [L 2-estimates for the ∂̄-operator of a semipositive holomorphic vector bundle over a complete Kähler manifold]. Ann Sci École Norm Sup, 1982, 15: 457–511MathSciNetzbMATHGoogle Scholar
  10. 10.
    Demailly J-P. Fonction de Green pluricomplexe et mesures pluriharmoniques (in French) [Pluricomplex Green functions and pluriharmonic measures]. In: Séminaire de Théorie Spectrale et Géométrie, No. 4, Année 1985–1986. Univ Grenoble I: Saint-Martin-d’Hères, 1986, 131–143Google Scholar
  11. 11.
    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, September 1997. Progress in Mathematics, Birkhäuser, 2000Google Scholar
  12. 12.
    Demailly J-P. Complex Analytic and Differential Geometry. http://www-fourier.ujf-grenoble.fr/demailly/books.html
  13. 13.
    Demailly J-P. Kähler manifolds and transcendental techniques in algebraic geometry. In: Proceedings of the International Congress of Mathematicians 2006, vol. I. Zürich: Eur Math Soc, 2007, 153–186Google Scholar
  14. 14.
    Demailly J-P. Analytic Methods in Algebraic Geometry. Beijing: Higher Education Press, 2010Google Scholar
  15. 15.
    Deng F S, Guan Q A, Zhang L Y. Some properties of squeezing functions on bounded domains. Pacific J Math, 2012, 257: 319–341CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Farkas H M, Kra I. Riemann Surfaces, 2nd ed. Graduate Texts in Mathematics, 71. New York: Springer-Verlag, 1992CrossRefzbMATHGoogle Scholar
  17. 17.
    Fornæs J E, Narasimhan R. The Levi problem on complex spaces with singularities. Math Ann, 1980, 248: 47–72CrossRefMathSciNetGoogle Scholar
  18. 18.
    Guan Q A, Zhou X Y. Optimal constant problem in the L 2 extension theorem. C R Math Acad Sci Paris Ser I, 2012, 350: 753–756CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Guan Q A, Zhou X Y. Generalized L 2 extension theorem and a conjecture of Ohsawa. C R Acad Sci Paris Ser I, 2013, 351: 111–114CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    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 Ser I, 2011, 349: 797–800CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Gunning R C, Rossi H. Analytic Functions of Several Complex Variables. Englewood Cliffs, NJ: Prentice-Hall, 1965zbMATHGoogle Scholar
  22. 22.
    D’Hoker E, Phong D H. The geometry of string perturbation theory. Rev Modern Phys, 1988, 60: 917CrossRefMathSciNetGoogle Scholar
  23. 23.
    Hörmander L. L 2 estimates and existence theorems for the ∂̄ operater. Acta Math, 1965, 113: 89–152CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Hörmander L. An Introduction to Complex Analysis in Several Variables, 3rd ed. New York: Elsevier Science Publishing, 1990zbMATHGoogle Scholar
  25. 25.
    Manivel L. Un théorème de prolongement L 2 de sections holomorphes d’un fibré vectoriel. Math Z, 1993, 212: 107–122CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    McNeal J. On large values of L 2 holomorphic functions. Math Res Lett, 1996, 3: 247–259CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    McNeal J, Varolin D. Analytic inversion of adjunction: L 2 extension theorems with gain. Ann L’Inst Fourier (Grenoble), 2007, 57: 703–718CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Ohsawa T. On the extension of L 2 holomorphic functions, II. Publ Res Inst Math Sci, 1988, 24: 265–275CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Ohsawa T. On the extension of L 2 holomorphic functions, III: Negligible weights. Math Z, 1995, 219: 215–225CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Ohsawa T. On the extension of L 2 holomorphic functions, IV: A new density concept. In: Geometry and Analysis on Complex Manifolds. River Edge, NJ: World Sci Publ, 1994, 157–170CrossRefGoogle Scholar
  31. 31.
    Ohsawa T. Addendum to “On the Bergman kernel of hyperconvex domains”. Nagoya Math J, 1995, 137: 145–148MathSciNetzbMATHGoogle Scholar
  32. 32.
    Ohsawa T. On the extension of L 2 holomorphic functions, V: Effects of generalization. Nagoya Math J, 2001, 161: 1–21. Erratum to: “On the extension of L 2 holomorphic functions, V: Effects of generalization” [Nagoya Math J, 2001, 161: 1–21]. Nagoya Math J, 2001, 163: 229MathSciNetzbMATHGoogle Scholar
  33. 33.
    Ohsawa T, Takegoshi K. On the extension of L 2 holomorphic functions. Math Z, 1987, 195: 197–204CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Pommerenke C, Suita N. Capacities and Bergman kernels for Riemann surfaces and Fuchsian groups. J Math Soc Japan, 1984, 36: 637–642CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Sario L, Oikawa K. Capacity Functions. Die Grundlehren der Mathematischen Wissenschaften, Band 149. New York: Springer-Verlag, 1969CrossRefzbMATHGoogle Scholar
  36. 36.
    Siu Y-T. The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi. In: Geometric Complex Analysis (Hayama, 1995). River Edge, NJ: World Sci Publ, 1996, 577–592Google Scholar
  37. 37.
    Siu Y-T. Invariance of plurigenera. Invent Math, 1998, 134: 661–673CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    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. Berlin: Springer-Verlag, 2002, 223–277CrossRefGoogle Scholar
  39. 39.
    Siu Y-T. Some recent transcendental techniques in algebraic and complex geometry. In: Proceedings of the International Congress of Mathematicians, vol. I (Beijing, 2002). Beijing: Higher Education Press, 2002, 439–448Google Scholar
  40. 40.
    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. Adv Complex Anal Appl, 2. Dordrecht: Kluwer Acad Publ, 2004, 45–83CrossRefGoogle Scholar
  41. 41.
    Straube E. Lectures on the L 2-Sobolev Theory of the ∂̄-Neumann Problem. ESI Lectures in Mathematics and Physics. Zürich: Eur Math Soc, 2010Google Scholar
  42. 42.
    Suita N. Capacities and kernels on Riemann surfaces. Arch Ration Mech Anal, 1972, 46: 212–217CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Wentworth R. Precise constants in bosonization formulas on Riemann surfaces, I. Comm Math Phys, 2008, 282: 339–355CrossRefMathSciNetzbMATHGoogle Scholar
  44. 44.
    Yamada A. Topics related to reproducing kemels, theta functions and the Suita conjecture (in Japanese). In: The Theory of Reproducing Kernels and their Applications (in Japanese) (Kyoto, 1998). Sūrikaisekikenkyūsho Kōkyūroku, No. 1067 (1998), 39–47Google Scholar
  45. 45.
    Zhou X Y. Some results related to group actions in several complex variables. In: Proceedings of the International Congress of Mathematicians, vol. II (Beijing, 2002). Beijing: Higher Education Press, 2002, 743–753Google Scholar
  46. 46.
    Zhou X Y. Invariant holomorphic extensionin several complex variables. Sci China Ser A, 2006, 49: 1593–1598CrossRefMathSciNetzbMATHGoogle Scholar
  47. 47.
    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 Pure Appl, 2012, 97: 579–601CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchPeking UniversityBeijingChina
  2. 2.School of Mathematical SciencesPeking UniversityBeijingChina
  3. 3.Institute of Mathematics, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  4. 4.Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingChina

Personalised recommendations